A324305 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a single '1' in row 1.

1, 2, 0, 2, 9, 3, 18, 3, 9, 64, 48, 200, 96, 200, 48, 64, 625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625, 7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776, 117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649, 2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152

Offset: 1

Author

Paul D. Hanna, Feb 28 2019

Keywords

• nonn
• tabf

Examples

			E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2) and satisfies A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
Explicitly,
A(x,y) = x + (2*y^2 + 2)*x^2/2! + (9*y^4 + 3*y^3 + 18*y^2 + 3*y + 9)*x^3/3! + (64*y^6 + 48*y^5 + 200*y^4 + 96*y^3 + 200*y^2 + 48*y + 64)*x^4/4! + (625*y^8 + 750*y^7 + 2775*y^6 + 2280*y^5 + 4300*y^4 + 2280*y^3 + 2775*y^2 + 750*y + 625)*x^5/5! + (7776*y^10 + 12960*y^9 + 46440*y^8 + 53640*y^7 + 100584*y^6 + 81360*y^5 + 100584*y^4 + 53640*y^3 + 46440*y^2 + 12960*y + 7776)*x^6/6! + (117649*y^12 + 252105*y^11 + 909979*y^10 + 1337700*y^9 + 2594501*y^8 + 2753415*y^7 + 3604342*y^6 + 2753415*y^5 + 2594501*y^4 + 1337700*y^3 + 909979*y^2 + 252105*y + 117649)*x^7/7! + ...
Setting y = 1 yields an o.g.f. of A006013:
A(x,y=1) = x + 2*x^2 + 7*x^3 + 30*x^4 + 143*x^5 + 728*x^6 + 3876*x^7 + 21318*x^8 + 120175*x^9 + ... + binomial(3*n-2,n-1)/n * x^n + ...
TRIANGLE.
This triangle of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;
117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649;
2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152; ...
RELATED SERIES.
The e.g.f. may be defined by A(x,y) = Series_Reversion( x/G(x,y) )
where G(x,y) is the e.g.f. of A201949 and equals
G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2)
so that
G(x,y) = 1 + (1 + y^2)*x + (1 + y + 2*y^2 + y^3 + y^4)*x^2/2! + (1 + 3*y + 5*y^2 + 6*y^3 + 5*y^4 + 3*y^5 + y^6)*x^3/3! + (1 + 6*y + 15*y^2 + 24*y^3 + 28*y^4 + 24*y^5 + 15*y^6 + 6*y^7 + y^8)*x^4/4! + (1 + 10*y + 40*y^2 + 90*y^3 + 139*y^4 + 160*y^5 + 139*y^6 + 90*y^7 + 40*y^8 + 10*y^9 + y^10)*x^5/5! + ...
and G(x,y) = x / Series_Reversion( A(x,y) ).
RELATED TRIANGLE.
Triangle A201949 of coefficients in G(x,y) such that A(x/G(x,y),y) = x begins
1;
1, 0, 1;
1, 1, 2, 1, 1;
1, 3, 5, 6, 5, 3, 1;
1, 6, 15, 24, 28, 24, 15, 6, 1;
1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1;
1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1; ...
where the g.f. of row n is Product_{k=0..n-1} (1 + k*y + y^2) for n >= 0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..10000 terms in rows 1..100 of this triangle in flattened form.

Crossrefs

Cf. A201949, A324304.

Cf. A324956, A324958.

Programs

PARI

{T(n, k)=polcoeff(prod(j=0, n-2,  n + j*y + n*y^2), k, y)}
{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}

PARI

/* A(x,y) = Series_Reversion(x/G(x,y)) where G(x,y) = e.g.f. A201949 */
{T(n,k) = my(G=1,A=x);
G = sum(m=0,n, x^m/m! * prod(j=0,m-1, 1 + j*y + y^2) +x*O(x^n));
A = serreverse(x/G);
n!*polcoeff(polcoeff(A,n,x),k,y)}
{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}

Formula

GENERATING FUNCTIONS.

E.g.f.: A(x,y) = x/(1 - y*A(x,y))^(1/y + y).

E.g.f.: A(x,y) = Series_Reversion( x*(1 - x*y)^(1/y + y) ), wrt x.

E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)

E.g.f.: A(x,y) = Series_Reversion( x/G(x,y) ) such that A(x/G(x,y),y) = x, where G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2) is the e.g.f. of A201949.

PARTICULAR ARGUMENTS.

E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).

E.g.f. at y = 1: A(x,y=1) = x*G(x)^2, where G = 1 + x*G(x)^3 is the g.f. of A001764.

FORMULAS INVOLVING TERMS.

Row sums: Sum_{k=0..2*n-2} T(n,k) = (3*n-2)!/(2*n-1)! for n >= 1.

T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.

T(n,n-1) = A324304(n) for n >= 1.