A145088 -id:A145088

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A145085

Square table, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) ) = R(n+1,x)^(n+1) with R(n,0) = 1; that is, the logarithmic derivative of the e.g.f. of row n equals the e.g.f. of row n+1 to the n+1 power, for n>=0.

+10
11

1, 1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 17, 39, 1, 1, 5, 31, 151, 322, 1, 1, 6, 49, 373, 1901, 3723, 1, 1, 7, 71, 741, 6250, 31851, 57577, 1, 1, 8, 97, 1291, 15457, 136711, 680265, 1147188, 1, 1, 9, 127, 2059, 32186, 416661, 3740137, 17947631, 28557909, 1, 1, 10, 161, 3081, 59677, 1030491, 13908049, 124143598, 571101141, 866222535

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1080

FORMULA

Row e.g.f.s satisfy: R(n,x) = exp( Integral R(n+1,x)^(n+1) dx ).

Row e.g.f.s satisfy: R(n,x) = 1 + Integral R(n,x)*R(n+1,x)^(n+1) dx.

Row e.g.f.s satisfy: R'(n,x)/R(n,x) = R(n+1,x)^(n+1) with R(n,0) = 1.

EXAMPLE

Table begins:

1,1,2,7,39,322,3723,57577,1147188,28557909,866222535,31362744620,...;

1,1,3,17,151,1901,31851,680265,17947631,571101141,21507723971,...;

1,1,4,31,373,6250,136711,3740137,124143598,4887140221,224203589593,...;

1,1,5,49,741,15457,416661,13908049,557865765,26296627233,...;

1,1,6,71,1291,32186,1030491,40606281,1911466016,105145651821,...;

1,1,7,97,2059,59677,2211823,100479577,5431432483,341787359269,...;

1,1,8,127,3081,101746,4283511,220384585,13453788426,953539677861,...;

1,1,9,161,4393,162785,7672041,440897697,30000376553,2365207145121,...;

1,1,10,199,6031,247762,12921931,820341289,61561430380,5344379824933,...;

1,1,11,241,8031,362221,20710131,1439328361,118089834231,11194348009941,...;

1,1,12,287,10429,512282,31860423,2405825577,214232473478,22019097106029,..;

1,1,13,337,13261,704641,47357821,3860734705,370824076621,41076472798081,..;

1,1,14,391,16563,946570,68362971,5983992457,616668950808,73237232298621,..;

PROG

(PARI) {T(n, k)=local(A=vector(n+k+2, j, 1+j*x)); for(i=0, n+k+1, for(j=0, n+k, m=n+k+1-j; A[m]=exp(intformal(A[m+1]^m+x*O(x^k))))); k!*polcoeff(A[n+1], k, x)}

for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A145080; rows: A145086, A145087, A145088, A145089.

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 01 2008

EXTENSIONS

Entry corrected by Paul D. Hanna, Sep 22 2020

STATUS

approved

A145083

Row 3 of square table A145080.

+10
6

1, 3, 21, 243, 4029, 88491, 2450085, 82648611, 3313381293, 154912893243, 8322387603093, 507658268093811, 34817646211022301, 2662987196578490187, 225556061819586894597, 21030571231219899162435

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Let R(n,x) be the e.g.f. of row n of square table A145080, then the

e.g.f.s satisfy: R(n,x) = exp( n*Integral R(n+1,x) dx ) for n>=1.

LINKS

Table of n, a(n) for n=0..15.

FORMULA

E.g.f.: A(x) = R(3,x) = exp( 3*Integral R(4,x) dx ) where R(n,x) is the e.g.f. of row n of square table A145080.

E.g.f.: A(x) = G(x)^3 where G(x) is the e.g.f. of A145088, which is row 3 of square table A145085.

PROG

(PARI) a(n)=local(A=vector(n+4, j, 1+j*x)); for(i=0, n+3, for(j=0, n, m=n+3-j; A[m]=exp(m*intformal(A[m+1]+x*O(x^n))))); n!*polcoeff(A[3], n, x)

(PARI) a(n)=local(A=vector(n+4, j, 1+j*x)); for(i=0, n+3, for(j=0, n, m=n+3-j; A[m]=exp(intformal(A[m+1]^(m+1)+x*O(x^n))))); n!*polcoeff(A[3]^3, n, x)

(PARI) a(n)=local(A=1); for(k=0, n-1, A=exp(intformal((n-k+2)*(A+x*O(x^n))))); n!*polcoeff(A, n)

for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 08 2014

CROSSREFS

Cf. A145080, A145081, A145082, A145084, A145085, A145088.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 01 2008

STATUS

approved

A145086

Row 0 of square table A145085.

+10
6

1, 1, 2, 7, 39, 322, 3723, 57577, 1147188, 28557909, 866222535, 31362744620, 1332663774173, 65529062871157, 3684878011841690, 234605021214637355, 16766728751635089083, 1335146927494755758530, 117695398260381996143695

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Let S(n,x) be the e.g.f. of row n of square table A145085, then the e.g.f.s satisfy: S(n,x) = exp( Integral S(n+1,x)^(n+1) dx ) for n>=0.

LINKS

Paul D. Hanna, Table of n, a(n) for n=0..60

FORMULA

E.g.f.: A(x) = exp( Integral R(1,x) dx ) where R(1,x) is the e.g.f. of A145081, which is row 1 of square table A145080.

PROG

(PARI) {a(n)=local(A=vector(n+2, j, 1+j*x)); for(i=0, n+1, for(j=0, n, m=n+1-j; A[m]=exp(m*intformal(A[m+1]+x*O(x^n))))); n!*polcoeff(exp(intformal(A[1])), n, x)}

(PARI) {a(n)=local(A=vector(n+2, j, 1+j*x)); for(i=0, n+1, for(j=0, n, m=n+1-j; A[m]=exp(intformal(A[m+1]^(m+1)+x*O(x^n))))); n!*polcoeff(exp(intformal(A[1])), n, x)}

CROSSREFS

Cf. A145085, A145087, A145088, A145089; A145080, A145081.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 01 2008

STATUS

approved

A145087

Row 2 of square table A145085.

+10
5

1, 1, 4, 31, 373, 6250, 136711, 3740137, 124143598, 4887140221, 224203589593, 11819532185476, 707883494843341, 47708648339054629, 3589347850731252292, 299381557667730507907, 27518788652896695773041

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Let S(n,x) be the e.g.f. of row n of square table A145085, then the e.g.f.s satisfy: S(n,x) = exp( Integral S(n+1,x)^(n+1) dx ) for n>=0.

LINKS

Table of n, a(n) for n=0..16.

FORMULA

E.g.f.: A(x) = S(2,x) = exp( Integral S(3,x)^3 dx ) where S(n,x) is the e.g.f. of row n of square table A145085.

E.g.f.: A(x) = R(2,x)^(1/2) = exp( Integral R(3,x) dx ) where R(2,x) = e.g.f. of A145082 and R(3,x) = e.g.f. of A145083.

PROG

(PARI) {a(n)=local(A=vector(n+3, j, 1+j*x)); for(i=0, n+2, for(j=0, n, m=n+2-j; A[m]=exp(m*intformal(A[m+1]+x*O(x^n))))); n!*polcoeff(A[2]^(1/2), n, x)}

(PARI) {a(n)=local(A=vector(n+3, j, 1+j*x)); for(i=0, n+2, for(j=0, n, m=n+2-j; A[m]=exp(intformal(A[m+1]^(m+1)+x*O(x^n))))); n!*polcoeff(A[2], n, x)}

CROSSREFS

Cf. A145085, A145086, A145088, A145089; A145080, A145082.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 01 2008

STATUS

approved

A145089

Row 4 of square table A145085.

+10
5

1, 1, 6, 71, 1291, 32186, 1030491, 40606281, 1911466016, 105145651821, 6645220590851, 476096681256716, 38249611004598701, 3415114289928480181, 336324126216378275806, 36299781235381103548731

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Let S(n,x) be the e.g.f. of row n of square table A145085, then the e.g.f.s satisfy: S(n,x) = exp( Integral S(n+1,x)^(n+1) dx ) for n>=0.

LINKS

Table of n, a(n) for n=0..15.

FORMULA

E.g.f.: A(x) = S(4,x) = exp( Integral S(5,x)^5 dx ) where S(n,x) is the e.g.f. of row n of square table A145085.

E.g.f.: A(x) = R(4,x)^(1/4) = exp( Integral R(5,x) dx ) where R(4,x) = e.g.f. of A145084 and R(5,x) = e.g.f. of row 5 of square table A145080.

PROG

(PARI) {a(n)=local(A=vector(n+5, j, 1+j*x)); for(i=0, n+4, for(j=0, n, m=n+4-j; A[m]=exp(m*intformal(A[m+1]+x*O(x^n))))); n!*polcoeff(A[4]^(1/4), n, x)}

(PARI) {a(n)=local(A=vector(n+5, j, 1+j*x)); for(i=0, n+4, for(j=0, n, m=n+4-j; A[m]=exp(intformal(A[m+1]^(m+1)+x*O(x^n))))); n!*polcoeff(A[4], n, x)}

CROSSREFS

Cf. A145085, A145086, A145087, A145088; A145080, A145084.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 01 2008

STATUS

approved

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