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

0,3

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).

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.

Links

Table of n, a(n) for n=0..44.

Formula

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

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

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

Example

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.

Mathematica

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

Crossrefs

Cf. A000984, A007820, A129506.

Sequence in context: A069748 A064441 A110949 * A126343 A344951 A174539

Adjacent sequences: A226700 A226701 A226702 * A226704 A226705 A226706

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Jun 15 2013

Status

approved