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 A154229 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1), read by rows. 6
 1, 1, 1, 1, 38, 1, 1, 139, 139, 1, 1, 365, 8828, 365, 1, 1, 807, 70492, 70492, 807, 1, 1, 1592, 357459, 7062136, 357459, 1592, 1, 1, 2889, 1404923, 98777227, 98777227, 1404923, 2889, 1, 1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, 40, 280, 9560, 142600, 7780240, 200370080, 16155726160, ...}. The row sums of this class of sequences (see cross-references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = binomial(n+2, 2)^2 = A000537(n+1). - G. C. Greubel, Mar 02 2021 LINKS G. C. Greubel, Rows n = 0..50 of the triangle, flattened FORMULA T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1) with T(n, 0) = T(n, n) = 1. EXAMPLE Triangle begins as: 1; 1, 1; 1, 38, 1; 1, 139, 139, 1; 1, 365, 8828, 365, 1; 1, 807, 70492, 70492, 807, 1; 1, 1592, 357459, 7062136, 357459, 1592, 1; 1, 2889, 1404923, 98777227, 98777227, 1404923, 2889, 1; 1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 1; MAPLE T:= proc(n, k) option remember; if k=0 or k=n then 1 else T(n-1, k) + T(n-1, k-1) + binomial(n+2, 2)^2*T(n-2, k-1) fi; end: seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021 MATHEMATICA T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + Binomial[n+2, 2]^2*T[n-2, k-1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *) PROG (Sage) def f(n): return binomial(n+2, 2)^2 def T(n, k): if (k==0 or k==n): return 1 else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1) flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021 (Magma) f:= func< n | Binomial(n+2, 2)^2 >; function T(n, k) if k eq 0 or k eq n then return 1; else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1); end if; return T; end function; [T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021 CROSSREFS Cf. A154227, A154228, A154230, A154231, A154233. Cf. A000537. Sequence in context: A023930 A022072 A351221 * A225433 A225398 A037936 Adjacent sequences: A154226 A154227 A154228 * A154230 A154231 A154232 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Jan 05 2009 EXTENSIONS Edited by G. C. Greubel, Mar 02 2021 STATUS approved

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Last modified July 13 12:04 EDT 2024. Contains 374282 sequences. (Running on oeis4.)