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

1, 0, 1, 0, 30, 1, 0, 150, 240, 1, 0, 240, 6810, 1020, 1, 0, 120, 63540, 94890, 3120, 1, 0, 0, 271170, 2615340, 740640, 7770, 1, 0, 0, 604800, 32186070, 47271840, 4029690, 16800, 1, 0, 0, 730800, 214628400, 1281612570, 518276640, 17075940, 32760, 1, 0, 0, 453600, 859992000, 18459063000, 26947757970, 4027831080, 60171300, 59040, 1

Offset

0,5

Comments

Triangle here T_5(n,m) is such that C(n,m)^5 = Sum_{j=0..n} C(n,j)*T_5(j,m).

Equivalently, lower triangular matrix T_5 such that

|| C(n,m)^5 || = P * T_5 = A007318 * T_5.

T_5(n,m) = 0 for n < m and for 5*m < n.

Refer to comment to A262704.

Example:

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(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).

Links

Table of n, a(n) for n=0..54.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

Formula

T_5(n,m) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(j,m)^5.

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).

Example

Triangle starts:
[1];
[0,   1];
[0,  30,      1];
[0, 150,    240,       1];
[0, 240,   6810,    1020,      1];
[0, 120,  63540,   94890,   3120,    1];
[0,   0, 271170, 2615340, 740640, 7770, 1];

Mathematica

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 *)

Prog

(MuPAD)
// as a function
T_5:=(n, m)->_plus((-1)^(n-j)*binomial(n, j)*binomial(j, m)^5 $ j=0..n):
// as a matrix h x h
_P:=h->matrix([[binomial(n, m) $m=0..h]$n=0..h]):
_P_5:=h->matrix([[binomial(n, m)^5 $m=0..h]$n=0..h]):
_T_5:=h->_P(h)^-1*_P_5(h):
(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
(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

Crossrefs

Second diagonal (T_5(n+1,n)) is A061167(n+1).

Column T_5(n,2) is A122193(5,n).

Cf. A109983 (transpose of), A262704, A262705.

Cf. A078739, A078741.

Sequence in context: A030128 A137348 A137737 * A062513 A040906 A040907

Adjacent sequences: A262703 A262704 A262705 * A262707 A262708 A262709

Keyword

nonn,tabl,easy

Author

Giuliano Cabrele, Sep 30 2015

Status

approved