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A226703 Triangle read by rows: T(n,k) = binomial(2*n,k)*Stirling2(2*n-k,n). 0

%I #16 Feb 01 2018 10:32:40

%S 1,1,2,7,12,6,90,150,90,20,1701,2800,1820,560,70,42525,69510,47250,

%T 16800,3150,252,1323652,2153844,1506582,582120,131670,16632,924,

%U 49329280,80015936,57093036,23291268,5885880,924924,84084,3432,2141764053,3466045440,2509478400,1063782720,289429140,51891840,6006000,411840,12870

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

%C Polynomials based on Extended Tepper's Identity

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

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

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

%D G. P. Egorychev. “Integral Representation and the Computation of Combinatorial Sums.” Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, (1984).

%D F. J. Papp. “Another Proof of Tepper’s Inequality.” Math. Magazine 45 (1972): 119-121.

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

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

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

%e 1,

%e 1 +2*x,

%e 7 +12*x +6*x^2,

%e 90 +150*x +90*x^2 +20*x^3,

%e 1701 +2800*x +1820*x^2 +560*x^3 +70*x^4.

%t Flatten[Table[Binomial[2n,k]StirlingS2[2n-k,n],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Jun 19 2013 *)

%Y Cf. A000984, A007820, A129506.

%K nonn,tabl

%O 0,3

%A _Vladimir Kruchinin_, Jun 15 2013

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)