OFFSET
0,2
LINKS
Imad Eddine Bousbaa, Ali Chouria and Jean-Gabriel Luque, A combinatorial Hopf algebra for the boson normal ordering problem, arXiv:1512.05937 [math.CO], 2015.
FORMULA
T(0,0) = 1, and for p,q > 0, T(p,q) = Sum_{i=1..p} Sum_{j=0..i} Sum_{k=0..i} Sum_{l=0..j} l! binomial(j,l) binomial(q-k+l,l) binomial(i,j) binomial(i,k) T(p-i,q-k+l).
EXAMPLE
Triangle starts:
1;
2, 2;
10, 18, 8;
62, 154, 124, 32;
462, 1426, 1596, 760, 128;
3982, 14506, 20380, 13680, 4336, 512;
38646, 161042, 269284, 229448, 104032, 23520, 2048;
...
MAPLE
dpq := proc (p::integer, q::integer)::integer; local i, j, k, l, s; option remember; s := 0; if q < 0 or p < 0 or p < q then return 0 elif p = 0 and q = 0 then return 1 else for i to p do for j from 0 to i do for k from 0 to i do for l from 0 to j do s := s+factorial(l)*binomial(j, l)*binomial(q-k+l, l)*binomial(i, j)*binomial(i, k)*dpq(p-i, q-k+l) end do end do end do end do; return s end if end proc
MATHEMATICA
T[0, 0] = 1; T[p_, q_] := T[p, q] = Sum[l! Binomial[j, l] Binomial[q-k+l, l] Binomial[i, j] Binomial[i, k] T[p-i, q-k+l], {i, 1, p}, {j, 0, i}, {k, 0, i}, {l, 0, j}]; Table[T[p, q], {p, 0, 7}, {q, 0, p}] // Flatten (* Jean-François Alcover, Dec 21 2015 *)
PROG
(PARI) T(p, q) = if (!p && !q, 1, sum(i=1, p, sum(j=0, i, sum(k=0, i, sum(l=0, j, l!*binomial(j, l)*binomial(q-k+l, l)*binomial(i, j)*binomial(i, k)*T(p-i, q-k+l)))))); \\ Michel Marcus, Dec 21 2015
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ali Chouria, Dec 04 2015
EXTENSIONS
More terms from Michel Marcus, Dec 21 2015
STATUS
approved