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A262706 Triangle: Newton expansion of C(n,m)^5, read by rows.

%I A262706 #28 Sep 08 2022 08:46:14
%S A262706 1,0,1,0,30,1,0,150,240,1,0,240,6810,1020,1,0,120,63540,94890,3120,1,
%T A262706 0,0,271170,2615340,740640,7770,1,0,0,604800,32186070,47271840,
%U A262706 4029690,16800,1,0,0,730800,214628400,1281612570,518276640,17075940,32760,1,0,0,453600,859992000,18459063000,26947757970,4027831080,60171300,59040,1
%N A262706 Triangle: Newton expansion of C(n,m)^5, read by rows.
%C A262706 Triangle here T_5(n,m) is such that C(n,m)^5 = Sum_{j=0..n} C(n,j)*T_5(j,m).
%C A262706 Equivalently, lower triangular matrix T_5 such that
%C A262706 || C(n,m)^5 ||  =  P * T_5 = A007318 * T_5.
%C A262706 T_5(n,m) = 0 for n < m and for 5*m < n.
%C A262706 Refer to comment to A262704.
%C A262706 Example:
%C A262706 C(x,2)^5 = x^5*(x-1)^5/32 = 1*C(x,2) + 240*C(x,3) + 6810*C(x,4) + 63540*C(x,5) + 271170*C(x,6) + 604800*C(x,7) + 730800*C(x,8) + 453600*C(x,9) + 113400*C(x,10);
%C A262706 C(5,2)^5 = C(5,3)^5 = 100000 = 1*C(5,2) + 240*C(5,3) + 6810*C(5,4) + 63540*C(5,5) = 1*C(5,3) + 1020*C(5,4) + 94890*C(5,5).
%H A262706 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arxiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%F A262706 T_5(n,m) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(j,m)^5.
%F A262706 Also, let S(r,s)(n,m) denote the Generalized Stirling2 numbers as defined in the link above, then T_5(n,m) = n! / (m!)^5 * S(m,m)(5,n).
%e A262706 Triangle starts:
%e A262706 [1];
%e A262706 [0,   1];
%e A262706 [0,  30,      1];
%e A262706 [0, 150,    240,       1];
%e A262706 [0, 240,   6810,    1020,      1];
%e A262706 [0, 120,  63540,   94890,   3120,    1];
%e A262706 [0,   0, 271170, 2615340, 740640, 7770, 1];
%t A262706 T5[n_, m_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[j, m]^5, {j, 0, n}]; Table[T5[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 01 2015 *)
%o A262706 (MuPAD)
%o A262706 // as a function
%o A262706 T_5:=(n,m)->_plus((-1)^(n-j)*binomial(n,j)*binomial(j,m)^5 $ j=0..n):
%o A262706 // as a matrix h x h
%o A262706 _P:=h->matrix([[binomial(n,m) $m=0..h]$n=0..h]):
%o A262706 _P_5:=h->matrix([[binomial(n,m)^5 $m=0..h]$n=0..h]):
%o A262706 _T_5:=h->_P(h)^-1*_P_5(h):
%o A262706 (Magma) [&+[(-1)^(n-j)*Binomial(n,j)*Binomial(j,m)^5: j in [0..n]]: m in [0..n], n in [0..10]]; // _Bruno Berselli_, Oct 01 2015
%o A262706 (PARI) T_5(nmax) = {for(n=0, nmax, for(m=0, n, print1(sum(j=0, n, (-1)^(n-j)*binomial(n,j)*binomial(j,m)^5), ", ")); print())} \\ _Colin Barker_, Oct 01 2015
%Y A262706 Second diagonal (T_5(n+1,n)) is A061167(n+1).
%Y A262706 Column T_5(n,2) is A122193(5,n).
%Y A262706 Cf. A109983 (transpose of), A262704, A262705.
%Y A262706 Cf. A078739, A078741.
%K A262706 nonn,tabl,easy
%O A262706 0,5
%A A262706 _Giuliano Cabrele_, Sep 30 2015