A300280 Triangle defined by T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1), for n>=0, k = 0..n, as read by rows.

1, 0, 1, 0, 3, 1, 0, 5, 10, 1, 0, 7, 57, 21, 1, 0, 9, 252, 246, 36, 1, 0, 11, 969, 2158, 710, 55, 1, 0, 13, 3414, 15927, 10260, 1635, 78, 1, 0, 15, 11329, 104883, 122125, 35085, 3255, 105, 1, 0, 17, 35992, 637252, 1273192, 611130, 96992, 5852, 136, 1, 0, 19, 110625, 3647268, 12057412, 9199386, 2321004, 230972, 9756, 171, 1, 0, 21, 331298, 19935477, 106181320, 124315310, 47518716, 7261394, 492408, 15345, 210, 1, 0, 23, 971609, 105054633, 883422885, 1546241270, 865414802, 193797618, 19669302, 963795, 23045, 253, 1

Offset: 0

Author

Paul D. Hanna, Mar 01 2018

Keywords

• nonn
• tabl

Comments

Is there a closed-form expression for the terms T(n,k) of this triangle?

Row sums form A300279, with g.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).

Examples

			This triangle begins:
1;
0, 1;
0, 3, 1;
0, 5, 10, 1;
0, 7, 57, 21, 1;
0, 9, 252, 246, 36, 1;
0, 11, 969, 2158, 710, 55, 1;
0, 13, 3414, 15927, 10260, 1635, 78, 1;
0, 15, 11329, 104883, 122125, 35085, 3255, 105, 1;
0, 17, 35992, 637252, 1273192, 611130, 96992, 5852, 136, 1;
0, 19, 110625, 3647268, 12057412, 9199386, 2321004, 230972, 9756, 171, 1;
0, 21, 331298, 19935477, 106181320, 124315310, 47518716, 7261394, 492408, 15345, 210, 1;
0, 23, 971609, 105054633, 883422885, 1546241270, 865414802, 193797618, 19669302, 963795, 23045, 253, 1; ...
GENERATING FUNCTIONS.
G.f.: A(x,y) = Sum_{n>=0} x^n*y^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1).
Expanding,
G.f.: A(x,y) = 1 + x*y*(1+x)/(2 - (1+x))^2 + x^2*y^2*(1+x)^4/(2 - (1+x)^2)^3 + x^3*y^3*(1+x)^9/(2 - (1+x)^3)^4 + x^4*y^4*(1+x)^16/(2 - (1+x)^4)^5 + x^5*y^5*(1+x)^25/(2 - (1+x)^5)^6 + x^6*y^6*(1+x)^36/(2 - (1+x)^6)^7 + ...
Also, due to a series identity:
A(x,y) = 1/2 + (1 + x*y*(1+x))/2^2 + (1 + x*y*(1+x)^2)^2/2^3 + (1 + x*y*(1+x)^3)^3/2^4 + (1 + x*y*(1+x)^4)^4/2^5 + (1 + x*y*(1+x)^5)^5/2^6 + (1 + x*y*(1+x)^6)^6/2^7 + ... + (1 + x*y * (1+x)^n)^n / 2^(n+1) + ...
Explicitly,
G.f.: A(x,y) = 1 + y*x + (y^2 + 3*y)*x^2 + (y^3 + 10*y^2 + 5*y)*x^3 + (y^4 + 21*y^3 + 57*y^2 + 7*y)*x^4 + (y^5 + 36*y^4 + 246*y^3 + 252*y^2 + 9*y)*x^5 + (y^6 + 55*y^5 + 710*y^4 + 2158*y^3 + 969*y^2 + 11*y)*x^6 + (y^7 + 78*y^6 + 1635*y^5 + 10260*y^4 + 15927*y^3 + 3414*y^2 + 13*y)*x^7 + (y^8 + 105*y^7 + 3255*y^6 + 35085*y^5 + 122125*y^4 + 104883*y^3 + 11329*y^2 + 15*y)*x^8 + ...
The row sums begin
A300279 = [1, 1, 4, 16, 86, 544, 3904, 31328, 276798, 2660564, ...],
and has g.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).
RELATED TRIANGLE.
The coefficients in 1/A(x,y) forms the triangle:
1;
0, -1;
0, -3, 0;
0, -5, -4, 0;
0, -7, -38, -4, 0;
0, -9, -208, -104, -4, 0;
0, -11, -884, -1336, -202, -4, 0;
0, -13, -3268, -12112, -4768, -332, -4, 0;
0, -15, -11098, -89540, -75532, -12520, -494, -4, 0; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1080, as a flattened triangle read by Rows 0..45

Crossrefs

Cf. A300279 (row sums).

Programs

PARI

/* Must set N to a large value for accuracy: */ N=10000;
{T(n,k) = round( sum(j=0,N, binomial(j+k,k) * binomial((j+k)*k,n-k) / 2^(j+k+1)*1. ) )}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

PARI

/* Faster, without precision errors: */
{T(n,k) = my(A = sum(m=0, n, x^m * y^m * (1+x + x*O(x^n))^(m^2) / (2 - (1+x + x*O(x^n))^m )^(m+1) )); polcoeff(polcoeff(A, n,x), k,y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

Formula

T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1).

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n * y^k is given by:

(1) A(x,y) = Sum_{n>=0} (1 + x*y * (1+x)^n)^n / 2^(n+1).

(2) A(x,y) = Sum_{n>=0} x^n * y^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1).