A262705 Triangle: Newton expansion of C(n,m)^4, read by rows.

1, 0, 1, 0, 14, 1, 0, 36, 78, 1, 0, 24, 978, 252, 1, 0, 0, 4320, 8730, 620, 1, 0, 0, 8460, 103820, 46890, 1290, 1, 0, 0, 7560, 581700, 1159340, 185430, 2394, 1, 0, 0, 2520, 1767360, 13387570, 8314880, 595476, 4088, 1, 0, 0, 0, 3087000, 85806000, 170429490, 44341584, 1642788, 6552, 1

Offset

0,5

Comments

Triangle here T_4(n,m) is such that C(n,m)^4 = Sum_{j=0..n} C(n,j)*T_4(j,m).

Equivalently, lower triangular matrix T_4 such that

|| C(n,m)^4 || = A202750 = P * T_4 = A007318 * T_4.

T_4(n,m) = 0 for n < m and for 4*m < n.

Refer to comment to A262704.

Example:

C(x,2)^4 = x^4*(x-1)^4 /16 = 1*C(x,2) + 78*C(x,3) + 978*C(x,4) + 4320*C(x,5) + 8460*C(x,6) + 7560*C(x,7) + 2520*C(x,8);

C(5,2)^4 = C(5,3)^4 = 10000 = 1*C(5,2) + 78*C(5,3) + 978*C(5,4) + 4320*C(5,5) = 1*C(5,3) + 252*C(5,4) + 8730*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_4(n,m) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(j,m)^4.

Also, let S(r,s)(n,m) denote the Generalized Stirling2 numbers as defined in the link above, then T_4(n,m) = n! / (m!)^4 * S(m,m)(4,n).

Example

Triangle starts:
[1];
[0,  1];
[0, 14,    1];
[0, 36,   78,      1];
[0, 24,  978,    252,     1];
[0,  0, 4320,   8730,   620,    1];
[0,  0, 8460, 103820, 46890, 1290, 1];

Mathematica

T4[n_, m_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[j, m]^4, {j, 0, n}]; Table[T4[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 01 2015 *)

Prog

(MuPAD)
// as a function
T_4:=(n, m)->_plus((-1)^(n-j)*binomial(n, j)*binomial(j, m)^4 $ j=0..n):
// as a matrix h x h
_P:=h->matrix([[binomial(n, m) $m=0..h]$n=0..h]):
_P_4:=h->matrix([[binomial(n, m)^4 $m=0..h]$n=0..h]):
_T_4:=h->_P(h)^-1*_P_4(h):
(Magma) [&+[(-1)^(n-j)*Binomial(n, j)*Binomial(j, m)^4: j in [0..n]]: m in [0..n], n in [0..10]]; // Bruno Berselli, Oct 01 2015
(PARI) T_4(nmax) = {for(n=0, nmax, for(m=0, n, print1(sum(j=0, n, (-1)^(n-j)*binomial(n, j)*binomial(j, m)^4), ", ")); print())} \\ Colin Barker, Oct 01 2015

Crossrefs

Row sums are, by definition, the inverse binomial transform of A005260.

Second diagonal (T_4(n+1,n)) is A058895(n+1).

Column T_4(n,2) is A122193(4,n).

Cf. A109983 (transpose of), A262704, A262706.

Cf. A078739, A078741.

Sequence in context: A228304 A002393 A185284 * A232210 A040199 A173747

Adjacent sequences: A262702 A262703 A262704 * A262706 A262707 A262708

Keyword

nonn,tabl,easy

Author

Giuliano Cabrele, Sep 30 2015

Status

approved