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

0,5

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).

Links

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

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).

Example

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; ...

Prog

(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(""))

Crossrefs

Cf. A300279 (row sums).

Sequence in context: A201667 A175779 A280819 * A376727 A135670 A096754

Adjacent sequences: A300277 A300278 A300279 * A300281 A300282 A300283

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 01 2018

Status

approved