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 A157155 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4, read by rows. 23
 1, 1, 1, 1, 6, 1, 1, 31, 31, 1, 1, 156, 462, 156, 1, 1, 781, 5442, 5442, 781, 1, 1, 3906, 57263, 124860, 57263, 3906, 1, 1, 19531, 566153, 2335435, 2335435, 566153, 19531, 1, 1, 97656, 5396164, 38814088, 71413750, 38814088, 5396164, 97656, 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, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4. T(n, n-k, m) = T(n, k, m). T(n, 1, 4) = A003463(n). - G. C. Greubel, Jan 10 2022 EXAMPLE Triangle begins as: 1; 1, 1; 1, 6, 1; 1, 31, 31, 1; 1, 156, 462, 156, 1; 1, 781, 5442, 5442, 781, 1; 1, 3906, 57263, 124860, 57263, 3906, 1; 1, 19531, 566153, 2335435, 2335435, 566153, 19531, 1; 1, 97656, 5396164, 38814088, 71413750, 38814088, 5396164, 97656, 1; MATHEMATICA T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m] + (m*k+1)*T[n-1, k, m] - m*k*(n-k)*T[n-2, k-1, m]]; Table[T[n, k, 4], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *) PROG (Sage) @CachedFunction def T(n, k, m): # A157155 if (k==0 or k==n): return 1 else: return (m*(n-k) +1)*T(n-1, k-1, m) + (m*k+1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) flatten([[T(n, k, 4) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 10 2022 CROSSREFS Cf. A007318 (m=0), A157152 (m=1), A157153 (m=2), A157154 (m=3), this sequence (m=4), A157156 (m=5). Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275. Cf. A003463. Sequence in context: A111578 A166349 A176429 * A022169 A156601 A178232 Adjacent sequences: A157152 A157153 A157154 * A157156 A157157 A157158 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 24 2009 EXTENSIONS Edited by G. C. Greubel, Jan 10 2022 STATUS approved

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Last modified July 15 02:08 EDT 2024. Contains 374323 sequences. (Running on oeis4.)