A226703 Triangle read by rows: T(n,k) = binomial(2*n,k)*Stirling2(2*n-k,n).

1, 1, 2, 7, 12, 6, 90, 150, 90, 20, 1701, 2800, 1820, 560, 70, 42525, 69510, 47250, 16800, 3150, 252, 1323652, 2153844, 1506582, 582120, 131670, 16632, 924, 49329280, 80015936, 57093036, 23291268, 5885880, 924924, 84084, 3432, 2141764053, 3466045440, 2509478400, 1063782720, 289429140, 51891840, 6006000, 411840, 12870

Offset: 0

Author

Vladimir Kruchinin, Jun 15 2013

Keywords

• nonn
• tabl

Comments

Polynomials based on Extended Tepper's Identity

P(n,x)=sum(j=0..n, (-1)^(n-j)*binomial(n,j)*(x+j)^(2*n))/n!.

P(n,x)=sum(j=0..n, binomial(2*n,j)*stirling2(2*n-j,n)*x^j).

P(n,1)=A129506(n).

Examples

			1,
1 +2*x,
7 +12*x +6*x^2,
90 +150*x +90*x^2 +20*x^3,
1701 +2800*x +1820*x^2 +560*x^3 +70*x^4.
		

References

• G. P. Egorychev. “Integral Representation and the Computation of Combinatorial Sums.” Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, (1984).
• F. J. Papp. “Another Proof of Tepper’s Inequality.” Math. Magazine 45 (1972): 119-121.

Crossrefs

Cf. A000984, A007820, A129506.

Programs

Mathematica

Flatten[Table[Binomial[2n,k]StirlingS2[2n-k,n],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 19 2013 *)

Formula

T(n,k) = binomial(2*n,k)*stirling2(2*n-k,n).

T(n,n) = A000984(n).

T(n,0) = A007820(n).