A361540 Expansion of e.g.f. A(x,y) satisfying A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n!, as a triangle read by rows.

1, 1, 1, 3, 4, 1, 22, 39, 18, 1, 269, 604, 426, 92, 1, 4616, 12625, 12040, 4550, 520, 1, 102847, 332766, 401355, 218300, 50085, 3222, 1, 2824816, 10574725, 15456756, 11017895, 3867080, 577731, 21700, 1, 92355769, 393171416, 676130644, 597596216, 284455150, 69038984, 7022596, 157544, 1

Offset

0,4

Comments

A202999(n) = Sum_{k=0..n} T(n,k).

A361053(n) = Sum_{k=0..n} T(n,k) * 2^k.

A361054(n) = Sum_{k=0..n} T(n,k) * 3^k.

A361055(n) = Sum_{k=0..n} T(n,k) * 4^k.

A361056(n) = Sum_{k=0..n} T(n,k) * 2^(n-k).

A361057(n) = Sum_{k=0..n} T(n,k) * 3^(n-k).

A203013(n) = Sum_{k=0..n} T(n,k) * 2^(n-k) * (-1)^k.

A155806(n) = T(n,0) for n >= 0; e.g.f. G(x) = Sum_{n>=0} G(x)^(n^2)*x^n/n!.

A361544(n) = T(n,1) for n >= 1.

A361549(n) = T(n,2) for n >= 2.

A185298(n) = T(n,n-1) for n >= 1; e.g.f. x*exp(x)*exp(x*exp(x)).

A361539(n) = T(n,n-2) for n >= 2.

A361688(n) = T(2*n,n) / binomial(2*n,n) for n >= 0.

Links

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

Formula

E.g.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! may be defined as follows.

(1) A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n!.

(2) A(x,y) = Sum_{n>=0} A(x,y)^(n^2) * exp(y*x*A(x,y)^n) * x^n/n!.

Example

E.g.f. A(x,y) = 1 + (y + 1)*x + (y^2 + 4*y + 3)*x^2/2! + (y^3 + 18*y^2 + 39*y + 22)*x^3/3! + (y^4 + 92*y^3 + 426*y^2 + 604*y + 269)*x^4/4! + (y^5 + 520*y^4 + 4550*y^3 + 12040*y^2 + 12625*y + 4616)*x^5/5! + (y^6 + 3222*y^5 + 50085*y^4 + 218300*y^3 + 401355*y^2 + 332766*y + 102847)*x^6/6! + (y^7 + 21700*y^6 + 577731*y^5 + 3867080*y^4 + 11017895*y^3 + 15456756*y^2 + 10574725*y + 2824816)*x^7/7! + (y^8 + 157544*y^7 + 7022596*y^6 + 69038984*y^5 + 284455150*y^4 + 597596216*y^3 + 676130644*y^2 + 393171416*y + 92355769)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k in e.g.f. A(x,y) begins:
[1];
[1, 1];
[3, 4, 1];
[22, 39, 18, 1];
[269, 604, 426, 92, 1];
[4616, 12625, 12040, 4550, 520, 1];
[102847, 332766, 401355, 218300, 50085, 3222, 1];
[2824816, 10574725, 15456756, 11017895, 3867080, 577731, 21700, 1];
[92355769, 393171416, 676130644, 597596216, 284455150, 69038984, 7022596, 157544, 1];
[3506278528, 16744363569, 33151425840, 35028273756, 21134516256, 7193104758, 1262445744, 90148860, 1224576, 1]; ...
RELATED TABLE.
The elements of this triangle T(n,k) when divided by binomial(n,k) yields the related triangle:
[1];
[1, 1];
[3, 2, 1];
[22, 13, 6, 1];
[269, 151, 71, 23, 1];
[4616, 2525, 1204, 455, 104, 1];
[102847, 55461, 26757, 10915, 3339, 537, 1];
[2824816, 1510675, 736036, 314797, 110488, 27511, 3100, 1];
[92355769, 49146427, 24147523, 10671361, 4063645, 1232839, 250807, 19693, 1];
[3506278528, 1860484841, 920872940, 417003259, 167734256, 57088133, 15029116, 2504135, 136064, 1]; ...

Prog

(PARI) /* E.g.f. A(x, y) = Sum_{n>=0} (A(x, y)^n + y)^n * x^n/n! */
{T(n, k) = my(A = 1); for(i=1, n, A = sum(m=0, n, (A^m + y +x*O(x^n))^m * x^m/m! )); n!*polcoeff(polcoeff(A, n, x), k, y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(" "))
(PARI) /* E.g.f. A(x, y) = Sum_{n>=0} A(x, y)^(n^2) * exp(y*x*A(x, y)^n) * x^n/n! */
{T(n, k) = my(A=1); for(i=1, n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(y*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(polcoeff(A, n, x), k, y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(" "))

Crossrefs

Cf. A202999 (y=1), A361053 (y=2), A361054 (y=3), A361055 (y=4), A361056, A361057, A203013.

Cf. A155806 (T(n,0)), A361544 (T(n,1)), A361549 (T(n,2)), A185298 (T(n,n-1)), A361539 (T(n,n-2)), A361688 (T(2*n,n)/C(2*n,n)).

Sequence in context: A055133 A342186 A113084 * A354293 A255905 A055325

Adjacent sequences: A361537 A361538 A361539 * A361541 A361542 A361543

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 20 2023

Status

approved