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 A265199 Triangle read by rows: T(p,q) is the number of B-diagrams G such that omega(G)=p half edges and h_f(G)=q non-used outer non-cut half edges. 1
 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, 414654, 1935050, 3726220, 3811344, 2217392, 726496, 122816, 8192 (list; table; graph; refs; listen; history; text; internal format)
 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 Cf. A081294 (right diagonal), A266093 (row sums). Sequence in context: A316200 A179789 A334396 * A294755 A102446 A348957 Adjacent sequences:  A265196 A265197 A265198 * A265200 A265201 A265202 KEYWORD nonn,tabl AUTHOR Ali Chouria, Dec 04 2015 EXTENSIONS More terms from Michel Marcus, Dec 21 2015 STATUS approved

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Last modified May 26 04:43 EDT 2022. Contains 354074 sequences. (Running on oeis4.)