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 A155493 Triangle T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1), read by rows. 3
 1, 1, 1, 1, 15, 1, 1, 118, 118, 1, 1, 770, 3540, 770, 1, 1, 4671, 67810, 67810, 4671, 1, 1, 27321, 1039689, 3085355, 1039689, 27321, 1, 1, 156220, 14006244, 99524810, 99524810, 14006244, 156220, 1, 1, 878868, 173788752, 2602528824, 6090918372, 2602528824, 173788752, 878868, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS G. C. Greubel, Rows n = 0..50 of the triangle, flattened FORMULA T(n, k) = binomial(n+1, k)*t(n, k, m)/(k+1), where t(n,k,m) = (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m), t(n,1,m) = t(n,n,m) = 1, and m = 4. From G. C. Greubel, Apr 01 2022: (Start) T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1). T(n, n-k) = T(n, k). (End) EXAMPLE Triangle begins as: 1; 1, 1; 1, 15, 1; 1, 118, 118, 1; 1, 770, 3540, 770, 1; 1, 4671, 67810, 67810, 4671, 1; 1, 27321, 1039689, 3085355, 1039689, 27321, 1; 1, 156220, 14006244, 99524810, 99524810, 14006244, 156220, 1; 1, 878868, 173788752, 2602528824, 6090918372, 2602528824, 173788752, 878868, 1; MATHEMATICA t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1, k-1, m] + (m*k -(m -1))*t[n-1, k, m]]; T[n_, k_, m_]:= Binomial[n+1, k]*t[n+1, k+1, m]/(k+1); Table[T[n, k, 4], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 01 2022 *) PROG (Sage) @CachedFunction def t(n, k, m): if (k==1 or k==n): return 1 else: return (m*(n-k)+1)*t(n-1, k-1, m) + (m*k-m+1)*t(n-1, k, m) def T(n, k, m): return binomial(n+1, k)*t(n+1, k+1, m)/(k+1) flatten([[T(n, k, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 01 2022 CROSSREFS Cf. A001263 (m=0), A155467 (m=1), A155491 (m=3), this sequence (m=4). Cf. A142459. Sequence in context: A111805 A238754 A176226 * A156939 A174187 A174693 Adjacent sequences: A155490 A155491 A155492 * A155494 A155495 A155496 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Jan 23 2009 EXTENSIONS Edited by G. C. Greubel, Apr 01 2022 STATUS approved

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Last modified June 13 22:21 EDT 2024. Contains 373391 sequences. (Running on oeis4.)