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Search: a293129 -id:a293129
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A291937
G.f.: Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n.
+10
9
1, 2, 0, 4, -3, 6, -3, 8, -15, 28, -24, 12, 0, 14, -48, 96, -95, 18, 55, 20, -180, 232, -120, 24, -35, 76, -168, 460, -580, 30, 515, 32, -927, 804, -288, 456, -497, 38, -360, 1288, -1169, 42, 847, 44, -2958, 3700, -528, 48, -2599, 148, 2526, 2772, -5537, 54, 595, 5336, -6930, 3820, -840, 60, -791, 62, -960, 6448, -12351, 12936, -3167, 68, -15435, 6648, 21365, 72, -26646, 74, -1368, 35776, -23730, 8394, -16548, 80, 7101
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Compare o.g.f. to: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
Compare l.g.f. to: Sum_{n=-oo..+oo, n<>0} x^n * (1 - x^(n-1))^n / n = -log(1-x).
Whenever a(n+2) is a multiple of n > 7, then a(n+2)/n = -(n+4)/4, with very few exceptions (n = 18, 131, 412, ... and n = 10, a(12) = 0). In particular, when n-1 is a prime of the form p = 4k + 3, then a(p+3) = -(k+2)(p+1) (as compared to a(p) = p+1), except for k = 11, 16, 26, 31, 37, 41, .... What exactly are these exceptions? - M. F. Hasler, Oct 10 2017
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..6600
FORMULA
The o.g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n.
(2) A(x) = Sum_{n=-oo..+oo} n^2 * x^(2*n) * (1 - x^n)^(n-1).
(3) A(x) = Sum_{n=-oo..+oo} -n * x^(2*n) * (1 - x^n)^(n-1).
(4) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n * x^(n^2-n) / (1 - x^n)^n.
(5) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n^2 * x^(n^2-n) / (1 - x^n)^(n+1).
(6) A(x) = Limit_{k->oo} Sum_{n=-oo..+oo} x^(n-k) * (1 - x^n - x^(n+k))^n.
(7) A(x) = Limit_{k->oo} Sum_{n=-oo..+oo} x^(n-k) * (1 - x^n + n*x^(n+k))^n.
The l.g.f. L(x) = Sum_{n>=1} a(n) * x^n / n satisfies:
(8) L(x) = -1 + Sum_{n=-oo..+oo, n<>0} x^n * (1 - x^n)^n / n.
a(p) = p+1 for odd primes p.
EXAMPLE
O.g.f.: A(x) = 1 + 2*x + 4*x^3 - 3*x^4 + 6*x^5 - 3*x^6 + 8*x^7 - 15*x^8 + 28*x^9 - 24*x^10 + 12*x^11 + 14*x^13 - 48*x^14 + 96*x^15 - 95*x^16 + 18*x^17 + 55*x^18 + 20*x^19 - 180*x^20 + 232*x^21 - 120*x^22 + 24*x^23 - 35*x^24 + 76*x^25 - 168*x^26 + 460*x^27 - 580*x^28 + 30*x^29 + 515*x^30 +...
where A(x) = P(x) + Q(x) with
P(x) = x*(1-x) + 2*x^2*(1-x^2)^2 + 3*x^3*(1-x^3)^3 + 4*x^4*(1-x^4)^4 + 5*x^5*(1-x^5)^5 +...+ n * x^n * (1 + x^n)^n + ...
Q(x) = 1/(1-x) - 2*x^2/(1-x^2)^2 + 3*x^6/(1-x^3)^3 - 4*x^12/(1-x^4)^4 + 5*x^20/(1-x^5)^5 + ... + -(-1)^n * n * x^(n^2-n) / (1 - x^n)^n + ...
Explicitly,
P(x) = x + x^2 + 3*x^3 + 5*x^5 - x^6 + 7*x^7 - 8*x^8 + 18*x^9 - 15*x^10 + 11*x^11 - 3*x^12 + 13*x^13 - 35*x^14 + 65*x^15 - 64*x^16 + 17*x^17 + 27*x^18 + 19*x^19 - 126*x^20 + 168*x^21 - 99*x^22 + 23*x^23 - 16*x^24 + 50*x^25 - 143*x^26 + 351*x^27 - 413*x^28 + 29*x^29 + 340*x^30 + ...
Q(x) = 1 + x - x^2 + x^3 - 3*x^4 + x^5 - 2*x^6 + x^7 - 7*x^8 + 10*x^9 - 9*x^10 + x^11 + 3*x^12 + x^13 - 13*x^14 + 31*x^15 - 31*x^16 + x^17 + 28*x^18 + x^19 - 54*x^20 + 64*x^21 - 21*x^22 + x^23 - 19*x^24 + 26*x^25 - 25*x^26 + 109*x^27 - 167*x^28 + x^29 + 175*x^30 + ...
Also, A(x) = M(x) + N(x) with
M(x) = x^2 + 4*x^4*(1-x^2) + 9*x^6*(1-x^3)^2 + 16*x^8*(1-x^4)^3 + 25*x^10*(1-x^5)^4 + ... + n^2 * x^(2*n) * (1 - x^n)^(n-1) + ...
N(x) = 1/(1-x)^2 - 4*x^2/(1-x^2)^3 + 9*x^6/(1-x^3)^4 - 16*x^12/(1-x^4)^5 + 25*x^20/(1-x^5)^6 + ... + -(-1)^n * n^2 * x^(n^2-n) / (1 - x^n)^(n+1) + ...
Explicitly,
M(x) = x^2 + 4*x^4 + 5*x^6 + 16*x^8 - 18*x^9 + 25*x^10 - 3*x^12 + 49*x^14 - 100*x^15 + 112*x^16 - 99*x^18 + 234*x^20 - 294*x^21 + 121*x^22 + 56*x^24 - 100*x^25 + 169*x^26 - 648*x^27 + 931*x^28 - 1010*x^30 + ...
N(x) = 1 + 2*x - x^2 + 4*x^3 - 7*x^4 + 6*x^5 - 8*x^6 + 8*x^7 - 31*x^8 + 46*x^9 - 49*x^10 + 12*x^11 + 3*x^12 + 14*x^13 - 97*x^14 + 196*x^15 - 207*x^16 + 18*x^17 + 154*x^18 + 20*x^19 - 414*x^20 + 526*x^21 - 241*x^22 + 24*x^23 - 91*x^24 + 176*x^25 - 337*x^26 + 1108*x^27 - 1511*x^28 + 30*x^29 + 1525*x^30 + ...
Terms at powers of 2 begin:
a(2^n) = [2, 0, -3, -15, -95, -927, -12351, -457215, -137484287, -71927383551, -12774376215944191, -2073810501234874519551, -78004011261694477161745918353407, ...].
Terms at powers of 3 begin:
a(3^n) = [2, 4, 28, 460, 10774, 80195104, 2894790054826, ..., A292184(n), ...].
Terms at powers of 5 begin:
a(5^n) = [2, 6, 76, 379626, 1259880626, 4828768869002981409762696876, ...].
MATHEMATICA
terms = 200; Sum[n*x^n*(1 - x^n)^n, {n, -terms, terms}] + O[x]^terms //
CoefficientList[#, x]& (* Jean-François Alcover, Oct 11 2017 *)
PROG
(PARI) {a(n)=my(l=1+O(x^(2*n+2))); polcoeff(sum(k=-n-2, n+2, k*x^k*(l-x^k)^k), n)} \\ Edited by M. F. Hasler, Oct 11 2017
(PARI) {a(n) = my(l=1+O(x^(2*n+2))); polcoeff(sum(k=-n-2, n+2, if(k, k^2 * x^(2*k) * (l - x^k)^(k-1))), n)} \\ Edited by M. F. Hasler, Oct 11 2017
(PARI) {a(n) = my(x='x+O('x^(2*n+2))); polcoeff(sum(k=-n-2, sqrtint(2*n)+2, -(-1)^k * k * x^(k^2-k) / (1 - x^k)^k), n)} \\ Edited by M. F. Hasler, Oct 11 2017
(PARI) {a(n) = my(x='x+O('x^(2*n+2))); polcoeff( sum(k=-n-2, sqrtint(2*n), if(k, -(-1)^k * k * x^(k^2-k) / (1 - x^k)^(k+1) )), n)} \\ Edited by M. F. Hasler, Oct 11 2017
for(n=0, 80, print1(a(n), ", "))
(PARI) A291937_vec(n)={my(x='x+O('x^(2*n+2))); -Vec(sum(k=-n-2, sqrtint(2*n), if(k, (-1)^k*k*x^(k^2-k)/(1-x^k)^(k+1))))[1..n+1]} \\ In case several values in a(0..n) are required, it is most efficient to compute the whole vector at once. E.g., sum(n=0..150, a(n)) takes ~ 10 sec., vecsum(A291937_vec(150)) takes ~ 0.1 sec. - M. F. Hasler, Oct 11 2017
CROSSREFS
Cf. A292184, A262007, A292177, A293129.
KEYWORD
sign,look
AUTHOR
Paul D. Hanna, Sep 06 2017
STATUS
approved
A293597
L.g.f.: Sum_{n>=1} (x - x^(2*n-1))^(2*n-1) / (2*n-1).
+10
4
0, 1, -4, 8, -11, 1, 14, 1, -50, 58, 1, 1, -54, 1, -28, 311, -340, 1, 75, 1, -81, 345, -44, 1, -1427, 1531, -52, 496, -1253, 1, 1343, 1, -2924, 738, -68, 9444, -10073, 1, -76, 1028, 3691, 1, -4691, 1, -6941, 21295, -92, 1, -55580, 33034, 28180, 1752, -11479, 1, -54063, 42847, 19437, 2186, -116, 1, -77934, 1, -124, 238507, -169032, 85151, -188859, 1, -25755, 3198, 432636, 1, -513328, 1, -148, 157041, -36005, 711327, -465347, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..2050
EXAMPLE
L.g.f.: A(x) = x^3/3 - 4*x^5/5 + 8*x^7/7 - 11*x^9/9 + x^11/11 + 14*x^13/13 + x^15/15 - 50*x^17/17 + 58*x^19/19 + x^21/21 + x^23/23 - 54*x^25/25 + x^27/27 - 28*x^29/29 + 311*x^31/31 - 340*x^33/33 + x^35/35 + 75*x^37/37 + x^39/39 - 81*x^41/41 + 345*x^43/43 - 44*x^45/45 + x^47/47 - 1427*x^49/49 + 1531*x^51/51 - 52*x^53/53 + 496*x^55/55 - 1253*x^57/57 + x^59/59 + 1343*x^61/61 + x^63/63 - 2924*x^65/65 +...
which may be written as
A(x) = (x - x) + (x - x^3)^3/3 + (x - x^5)^5/5 + (x - x^7)^7/7 + (x - x^9)^9/9 + (x - x^11)^11/11 + (x - x^13)^13/13 + (x - x^15)^15/15 + (x - x^17)^17/17 + (x - x^19)^19/19 + (x - x^21)^21/21 +...+ (x - x^(2*n-1))^(2*n-1)/(2*n-1) +...
The coefficient of x^(2^n+1)/(2^n+1) in A(x) for n>=1 begins:
[1, -4, -11, -50, -340, -2924, -169032, -33445208, -21619038032, 1 - A293599(n), ...].
PROG
(PARI) {a(n) = my(A, Ox = O(x^(2*n+1)));
A = sum(m=1, n+1, (x - x^(2*m-1) +Ox)^(2*m-1) / (2*m-1) );
(2*n-1)*polcoeff(A, 2*n-1)}
for(n=1, 80, print1(a(n), ", "))
CROSSREFS
Cf. A293129, A293598, A293599.
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 12 2017
STATUS
approved
A293598
L.g.f.: Sum_{n>=1} x^((2*n-1)^2) / ( (2*n-1) * (1 - x^(2*n))^(2*n-1) ).
+10
4
1, 3, 5, 7, 12, 11, 26, 15, 51, 19, 91, 23, 155, 27, 232, 62, 341, 35, 592, 39, 656, 344, 870, 47, 1820, 51, 1431, 1441, 1843, 59, 4758, 63, 2925, 4489, 3197, 71, 11899, 75, 4466, 11376, 7650, 83, 23052, 87, 12816, 25025, 7936, 95, 57133, 99, 10706, 49131, 37220, 107, 79570, 2146, 62828, 89263, 15951, 119, 228096, 123, 19500, 152146, 169033, 18864, 218253, 135, 267972, 246308, 75153, 143, 724159, 147, 33227, 490146, 629034, 155, 512448, 159
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..2050
EXAMPLE
L.g.f.: A(x) = x + 3*x^3/3 + 5*x^5/5 + 7*x^7/7 + 12*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + 51*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + 341*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + 2925*x^65/65 +...
which may be written as
A(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...
The coefficient of x^(2^n+1)/(2^n+1) in A(x) for n>=1 begins:
[3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, ..., A293599(n), ...].
MATHEMATICA
nmax = 80; Table[(CoefficientList[Series[Sum[x^((2*k - 1)^2)/((2*k - 1)*(1 - x^(2*k))^(2*k - 1)), {k, 1, 2*nmax + 1}], {x, 0, 2*nmax + 1}], x] * Range[0, 2*nmax + 1])[[2*n]], {n, 1, nmax}] (* Vaclav Kotesovec, Oct 15 2017 *)
PROG
(PARI) {a(n) = my(A, Ox = O(x^(2*n+1)));
A = sum(m=1, sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );
(2*n-1)*polcoeff(A, 2*n-1)}
for(n=1, 80, print1(a(n), ", "))
CROSSREFS
Cf. A293129, A293597, A293599.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 12 2017
STATUS
approved
A293599
The coefficient of x^(2^n+1)/(2^n+1) in the l.g.f. of A293598 for n>=1.
+10
4
3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, 3270933679995185, 344648907850020294305, 20381496562418327375031168210529, 303229033555187108276527297692992345985345, 533360801574481336406792124161160375221861972273961952144925889, 331572178130571824652402094592695034861147899073590997231695381294750188182312600193
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The l.g.f. of A293598 is Sum_{n>=1} x^((2*n-1)^2)/((2*n-1)*(1 - x^(2*n))^(2*n-1)).
The coefficient of x^(2^n+1)/(2^n+1) in the l.g.f. of A293597 equals 1 - a(n) for n>=2.
What is the rate of growth of this sequence?
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..19
EXAMPLE
L.g.f. of A293598: Q(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...
Explicitly,
Q(x) = x + (3)*x^3/3 + (5)*x^5/5 + 7*x^7/7 + (12)*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + (51)*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + (341)*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + (2925)*x^65/65 +...
This sequence equals the coefficient of x^(2^n+1)/(2^n+1) in Q(x) for n>=1.
MATHEMATICA
nmax = 10; Table[(CoefficientList[Series[Sum[x^((2*k - 1)^2)/((2*k - 1)*(1 - x^(2*k))^(2*k - 1)), {k, 1, 2^nmax + 1}], {x, 0, 2^nmax + 1}], x] * Range[0, 2^nmax + 1])[[2^n + 2]], {n, 1, nmax}] (* Vaclav Kotesovec, Oct 15 2017 *)
PROG
(PARI) {A293598(n) = my(Q, Ox = O(x^(2*n+1)));
Q = sum(m=1, sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );
(2*n-1)*polcoeff(Q, 2*n-1)}
for(n=0, 15, print1(A293598(2^n+1), ", "))
CROSSREFS
Cf. A293129, A293597, A293598.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 12 2017
STATUS
approved
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