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Search: a212917 -id:a212917
Displaying 1-9 of 9 results found.
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A161630
E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).
+10
16
1, 1, 3, 19, 181, 2321, 37501, 731935, 16758393, 440525377, 13077834841, 432796650551, 15799794395749, 630773263606513, 27339525297079269, 1278550150117141231, 64171287394646697841, 3440711053857464325377
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..373
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265
FORMULA
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(n-1,n-k).
E.g.f. satisfies: A(x) = exp(x) * A(x)^(x*A(x)). - Paul D. Hanna, Aug 02 2013
a(n) ~ n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2) * exp(n) * c^(2*n+3/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014
EXAMPLE
E.g.f: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 + x^4*A(x)^4 +...
MATHEMATICA
Table[Sum[n! * (n-k+1)^(k-1)/k! * Binomial[n-1, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 10 2014 *)
PROG
(PARI) {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(n-k+m)^(k-1)*binomial(n-1, n-k)))}
(PARI) {a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(x/(1-x*A))); n!*polcoeff(A^m, n)}
CROSSREFS
Cf. A161633 (e.g.f. = log(A(x))/x).
Cf. A212722, A212917, A245265, A125500.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 17 2009
STATUS
approved
A212722
E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^2) ).
+10
12
1, 1, 3, 25, 313, 5341, 115651, 3036517, 93767185, 3330162073, 133737097411, 5992748728561, 296433923379529, 16044427276953973, 943207466055927619, 59848531677741706621, 4076826825898115406241, 296742863575079244130225
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..358
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265
FORMULA
a(n) = Sum_{k=0..n} n! * (1 + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(m + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ n^(n-1) * (1+1/(2*c))^(n+1/2) / (2*sqrt(1+c) * exp(n) * c^n), where c = LambertW(1/sqrt(2)) = 0.450600515864833072257... . - Vaclav Kotesovec, Jul 15 2014
EXAMPLE
E.g.f: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ...
such that, by definition:
log(A(x))/x = 1 + x*A(x)^2 + x^2*A(x)^4 + x^3*A(x)^6 + x^4*A(x)^8 + ...
Related expansions:
log(A(x)) = x/(1-x*A(x)^2) = x + 2*x^2/2! + 18*x^3/3! + 216*x^4/4! + 3640*x^5/5! + 78000*x^6/6! + 2032464*x^7/7! + 62400128*x^8/8! + ... + n*A366232(n-1)*x^n/n! + ...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 68*x^3/3! + 880*x^4/4! + 15312*x^5/5! + 336064*x^6/6! +...
A(x)^4 = 1 + 4*x + 24*x^2/2! + 232*x^3/3! + 3232*x^4/4! + 59104*x^5/5! + 1343296*x^6/6! +...
MATHEMATICA
Table[Sum[n! * (1 + 2*(n-k))^(k-1)/k! * Binomial[n-1, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 15 2014 *)
PROG
(PARI) {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+2*(n-k))^(k-1)*binomial(n-1, n-k)))}
(PARI) {a(n, m=1)=local(A=1+x); for(i=1, n, A=exp(x/(1-x*A^2+x*O(x^n)))); n!*polcoeff(A^m, n)}
for(n=0, 21, print1(a(n), ", "))
CROSSREFS
Cf. A161630, A212917, A245265.
Cf. A366232 (log).
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 25 2012
STATUS
approved
A361094
E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^3) - 1 ).
+10
8
1, 1, 9, 166, 4717, 182136, 8911549, 528571408, 36864033945, 2956595372416, 268116203622961, 27128338649300736, 3029974270053623941, 370289278173654092800, 49150116757136815109733, 7041536364582774222616576, 1083004122024520209576760369
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Winston de Greef, Table of n, a(n) for n = 0..322
FORMULA
a(n) = n! * Sum_{k=0..n} (3*n+1)^(k-1) * binomial(n-1,n-k)/k!.
a(n) ~ (5 + sqrt(21))^n * n^(n-1) / (3^(3/4) * 7^(1/4) * 2^n * exp((3 - sqrt(21))/6 + (5 - sqrt(21))*n/2)). - Vaclav Kotesovec, Mar 02 2023
MATHEMATICA
Table[n! * Sum[(3*n+1)^(k-1) * Binomial[n-1, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 02 2023 *)
PROG
(PARI) a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(n-1, n-k)/k!);
CROSSREFS
Cf. A052873, A361093, A361095, A361096, A361097.
Cf. A212917, A361066.
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 01 2023
STATUS
approved
A366233
Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(3*x*A(x)).
+10
7
1, 1, 8, 87, 1428, 31125, 847818, 27785205, 1065267864, 46802921769, 2319200977230, 127985909702409, 7785440359004916, 517616528753919933, 37344834374921549154, 2906043724955696034285, 242627026212699695954352, 21634774261037677172406609, 2052077586846349144929739542
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.
LINKS
Table of n, a(n) for n=0..18.
FORMULA
a(n) = n! * Sum{k=0..n} binomial(n+1, n-k)/(n+1) * 3^k * (n-k)^k / k!.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 3^k * (n-k)^k/k!.
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x) * exp(3*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(3*x)) ).
(3) A( x/(1 + x*exp(3*x)) ) = 1 + x*exp(3*x).
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-3)*x*A(x)) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-(n+1)*x*A(x)).
a(n) ~ 3^n * (1 + 2*LambertW(sqrt(3)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(3)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(3)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023
EXAMPLE
E.g.f.: A(x) = 1 + x + 8*x^2/2! + 87*x^3/3! + 1428*x^4/4! + 31125*x^5/5! + 847818*x^6/6! + 27785205*x^7/7! + 1065267864*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(3*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+2*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+1*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(-0*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-1*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-2*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-3*x*A(x))/6! + ...
and
A(x) = 1 + 4*1*4^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 4*2*5^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 4*3*6^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 4*4*7^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 4*5*8^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + 254701*x^6/6! + ... + A212917(n)*x^n/n! + ...
MATHEMATICA
nmax = 20; A[_] = 0; Do[A[x_] = 1 + x*A[x] * E^(3*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)
PROG
(PARI) /* a(n, m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 3^k * (n-k)^k/k!)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(3*x +O(x^(n+2)))) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Cf. A161633, A366232, A366234, A366235.
Cf. A365773 (dual), A212917 (exp(x*A(x)).
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 05 2023
STATUS
approved
A245265
E.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^4)).
+10
5
1, 1, 3, 37, 649, 15461, 471571, 17456041, 760880625, 38178439849, 2167446089251, 137359883836781, 9612722107574521, 736277501363180557, 61265207586681046131, 5503291392884323494961, 530778414439201798454881, 54706967800114521799571921, 6000952913613549583603208515
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then
r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),
A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),
a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265
FORMULA
a(n) = Sum_{k=0..n} n! * (1 + 4*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ n^(n-1) * (1+2*LambertW(1))^(n+1/2) / (exp(n) * (LambertW(1))^(2*n) * (4*sqrt(1+LambertW(1)))). - Vaclav Kotesovec, Jul 15 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ...
MATHEMATICA
Table[Sum[n! * (1 + 4*(n-k))^(k-1)/k! * Binomial[n-1, n-k], {k, 0, n}], {n, 0, 20}]
PROG
(PARI) for(n=0, 30, print1(sum(k=0, n, n!*(1 + 4*(n-k))^(k-1)/k!*binomial(n-1, n-k)), ", ")) \\ G. C. Greubel, Nov 17 2017
CROSSREFS
Cf. A161630 (p=1), A212722 (p=2), A212917 (p=3).
Cf. A030178.
Cf. A366234 (log).
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Jul 15 2014
STATUS
approved
A361090
E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)) ).
+10
4
1, 1, 3, 7, -11, -239, -179, 24991, 192025, -3955391, -89483399, 552615031, 46231717621, 254468241457, -26683006147979, -571848064714289, 14926049610344881, 825004339886219521, -2973711136010539535, -1134313888244827421465, -17734152216328857754739
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Winston de Greef, Table of n, a(n) for n = 0..399
FORMULA
a(n) = n! * Sum_{k=1..n} (-n+k+1)^(k-1) * binomial(n-1,n-k)/k! for n>0.
PROG
(PARI) a(n) = if(n==0, 1, n!*sum(k=1, n, (-n+k+1)^(k-1)*binomial(n-1, n-k)/k!));
CROSSREFS
Cf. A161630, A212722, A212917, A361091, A361092.
Cf. A361067, A361095.
KEYWORD
sign
AUTHOR
Seiichi Manyama, Mar 01 2023
STATUS
approved
A361091
E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^2) ).
+10
4
1, 1, 3, 1, -71, -19, 10051, 12349, -3185391, -9346247, 1797304771, 9717361721, -1582301193527, -13722004186331, 2000705907453891, 25552516703201461, -3432004488804778079, -60960914621687232271, 7660860906885122096515
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Winston de Greef, Table of n, a(n) for n = 0..380
FORMULA
a(n) = n! * Sum_{k=0..n} (-2*n+2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
PROG
(PARI) a(n) = n!*sum(k=0, n, (-2*n+2*k+1)^(k-1)*binomial(n-1, n-k)/k!);
CROSSREFS
Cf. A161630, A212722, A212917, A361090, A361092.
Cf. A361068, A361096.
KEYWORD
sign
AUTHOR
Seiichi Manyama, Mar 01 2023
STATUS
approved
A361092
E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^3) ).
+10
4
1, 1, 3, -5, -107, 1041, 20701, -440033, -8464455, 343190593, 5639857561, -423764450889, -4968055259771, 754544622295153, 3846355902999429, -1818148417882379729, 6637679490204153841, 5658469355898945338625, -84578525845602646639823
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Winston de Greef, Table of n, a(n) for n = 0..368
FORMULA
a(n) = n! * Sum_{k=0..n} (-3*n+3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
PROG
(PARI) a(n) = n!*sum(k=0, n, (-3*n+3*k+1)^(k-1)*binomial(n-1, n-k)/k!);
CROSSREFS
Cf. A161630, A212722, A212917, A361090, A361091.
Cf. A361069, A361097.
KEYWORD
sign
AUTHOR
Seiichi Manyama, Mar 01 2023
STATUS
approved
A363354
E.g.f. satisfies A(x) = exp(x * (1 + x * A(x)^3)).
+10
4
1, 1, 3, 25, 277, 4221, 81421, 1891429, 51638217, 1618907257, 57332786041, 2264047223241, 98641443498973, 4700569138096885, 243213757144477029, 13579261873673960941, 813757288951509415441, 52098716516012891238129, 3548972379593741013388657
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..18.
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
E.g.f.: exp( x - LambertW(-3*x^2*exp(3*x))/3 ).
a(n) = n! * Sum_{k=0..n} (3*n-3*k+1)^(k-1) * binomial(k,n-k)/k!.
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x^2*exp(3*x))/3)))
CROSSREFS
Cf. A125500, A143768, A363529.
Cf. A212917.
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 17 2023
STATUS
approved
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