A245111 G.f.: A(x,y) = Sum_{n>=0} exp(-y/(1-n*x)) * y^n/(1-n*x)^n / n!.

1, 0, 1, 0, 1, 3, 0, 1, 12, 10, 0, 1, 35, 90, 35, 0, 1, 90, 525, 560, 126, 0, 1, 217, 2520, 5460, 3150, 462, 0, 1, 504, 10836, 42000, 46200, 16632, 1716, 0, 1, 1143, 43470, 280665, 519750, 342342, 84084, 6435, 0, 1, 2550, 166375, 1709400, 4969965, 5297292, 2312310, 411840, 24310

Offset

0,6

Comments

Compare g.f. to: 1/(1-x*y) = Sum_{n>=0} exp(-y*(1+n*x)) * y^n*(1+n*x)^n / n!.

Row sums equal A245110.

Antidiagonal sums: A218667.

Main diagonal is: C(2*n-1,n) (A001700).

Secondary diagonal: C(2*n-1,n)*n^2 (A002544).

Links

Paul D. Hanna, Table of n, a(n), of flattened triangle for rows 0..32

Formula

T(n,k) = Stirling2(n, k) * binomial(n+k-1, k-1) for k>0, where Stirling2(n,k) = A048993(n,k).

Example

G.f.: A(x,y) = 1 + x*y + x^2*(y + 3*y^2)
+ x^3*(y + 12*y^2 + 10*y^3)
+ x^4*(y + 35*y^2 + 90*y^3 + 35*y^4)
+ x^5*(y + 90*y^2 + 525*y^3 + 560*y^4 + 126*y^5)
+ x^6*(y + 217*y^2 + 2520*y^3 + 5460*y^4 + 3150*y^5 + 462*y^6) +...
where
A(x,y) = exp(-y) + exp(-y/(1-x))*y/(1-x) + (exp(-y/(1-2*x))*y^2/(1-2*x)^2)/2!
+ (exp(-y/(1-3*x))*y^3/(1-3*x)^3)/3! + (exp(-y/(1-4*x))*y^4/(1-4*x)^4)/4!
+ (exp(-y/(1-5*x))*y^5/(1-5*x)^5)/5! + (exp(-y/(1-6*x))*y^6/(1-6*x)^6)/6!
+ (exp(-y/(1-7*x))*y^7/(1-7*x)^7)/7! + (exp(-y/(1-8*x))*y^8/(1-8*x)^8)/8! +...
simplifies to a power series with only integer coefficients of x^n*y^k.
Triangle begins:
1;
0, 1;
0, 1, 3;
0, 1, 12, 10;
0, 1, 35, 90, 35;
0, 1, 90, 525, 560, 126;
0, 1, 217, 2520, 5460, 3150, 462;
0, 1, 504, 10836, 42000, 46200, 16632, 1716;
0, 1, 1143, 43470, 280665, 519750, 342342, 84084, 6435;
0, 1, 2550, 166375, 1709400, 4969965, 5297292, 2312310, 411840, 24310;
0, 1, 5621, 615780, 9754030, 42567525, 68549481, 47087040, 14586000, 1969110, 92378; ...
where T(n,k) = A048993(n,k) * C(n+k-1, k-1) for k>0.

Prog

(PARI) /* From definition: */
{T(n, k)=local(A=1+x*y); A=sum(k=0, n, 1/(1-k*x+x*O(x^n))^k*y^k/k!*exp(-y/(1-k*x+x*O(x^n))+y*O(y^n))); polcoeff(polcoeff(A, n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* From T(n, k) = Stirling2(n, k) * C(n+k-1, k-1) */
{Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{T(n, k)=if(k==0, 0^n, Stirling2(n, k) * binomial(n+k-1, k-1))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A245110, A218667, A001700, A002544, A048993.

Sequence in context: A145881 A232223 A380191 * A135313 A322670 A277410

Adjacent sequences: A245108 A245109 A245110 * A245112 A245113 A245114

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 12 2014

Status

approved