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

Author

Paul D. Hanna, Mar 20 2023

Keywords

• nonn
• tabl

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.

Examples

			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]; ...
		

Links

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

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)).

Programs

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(" "))

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!.