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 A118354 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118351. 6
 1, 1, 0, 1, 1, 0, 1, 2, 6, 0, 1, 3, 13, 42, 0, 1, 4, 21, 96, 325, 0, 1, 5, 30, 163, 770, 2688, 0, 1, 6, 40, 244, 1353, 6530, 23286, 0, 1, 7, 51, 340, 2093, 11760, 57612, 208659, 0, 1, 8, 63, 452, 3010, 18636, 105681, 523446, 1918314, 0, 1, 9, 76, 581, 4125, 27441, 170580, 973953, 4864795, 17994264, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS A118351 equals the central terms of pendular triangle A118350 and the lower diagonals of this triangle form the semi-diagonals of the triangle A118350. LINKS G. C. Greubel, Rows n = 0..100 of the triangle, flattened FORMULA Since g.f. G=G(x) of A118351 satisfies: G = 1 - 3*x*G + 3*x*G^2 + x*G^3 then T(n,k) = T(n-1,k) - 3*T(n-1,k-1) + 3*T(n,k-1) + T(n+1,k-1). Recurrence involving antidiagonals: T(n,k) = T(n-1,k) + Sum_{j=1..k} [4*T(n-1+j,k-j) - 3*T(n-2+j,k-j)] for n>k>=0. EXAMPLE Show: T(n,k) = T(n-1,k) - 3*T(n-1,k-1) + 3*T(n,k-1) + T(n+1,k-1) at n=8,k=4: T(8,4) = T(7,4) - 3*T(7,3) + 3*T(8,3) + T(9,3) or: 2093 = 1353 - 3*244 + 3*340 + 452. Triangle begins:   1;   1, 0;   1, 1,  0;   1, 2,  6,   0;   1, 3, 13,  42,    0;   1, 4, 21,  96,  325,     0;   1, 5, 30, 163,  770,  2688,      0;   1, 6, 40, 244, 1353,  6530,  23286,      0;   1, 7, 51, 340, 2093, 11760,  57612, 208659,       0;   1, 8, 63, 452, 3010, 18636, 105681, 523446, 1918314,        0;   1, 9, 76, 581, 4125, 27441, 170580, 973953, 4864795, 17994264, 0; ... MATHEMATICA T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==n, 0, T[n-1, k] -3*T[n-1, k-1] +3*T[n, k-1] +T[n+1, k-1]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2021 *) PROG (PARI) {T(n, k)=polcoeff((serreverse(x*(1-3*x+sqrt((1-3*x)*(1-7*x)+x*O(x^k)))/2/(1-3*x))/x)^(n-k), k)} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) (Sage) @CachedFunction def T(n, k):     if (k==0): return 1     elif (k==n): return 0     else: return T(n-1, k) - 3*T(n-1, k-1) + 3*T(n, k-1) + T(n+1, k-1) flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 18 2021 CROSSREFS Cf. A118350, A118351, A118352, A118353. Row sums: A151616. Sequence in context: A330327 A039907 A072340 * A080730 A232178 A016590 Adjacent sequences:  A118351 A118352 A118353 * A118355 A118356 A118357 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Apr 26 2006 STATUS approved

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Last modified September 17 06:16 EDT 2021. Contains 347478 sequences. (Running on oeis4.)