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 A135229 Triangle A000012(signed) * A103451 * A007318, read by rows. 2
 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 3, 6, 7, 4, 1, 1, 3, 9, 13, 11, 5, 1, 1, 4, 12, 22, 24, 16, 6, 1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS row sums = A005578 starting (1, 2, 3, 6, 11, 22, 43, 86, ...). LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened FORMULA T(n,k) = A000012(signed) * A103451 * A007318 as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; ...). T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = 1. - G. C. Greubel, Nov 20 2019 EXAMPLE First few rows of the triangle are: 1; 1, 1; 1, 1, 1; 1, 2, 2, 1; 1, 2, 4, 3, 1; 1, 3, 6, 7, 4, 1; 1, 3, 9, 13, 11, 5, 1; 1, 4, 12, 22, 24, 16, 6, 1; 1, 4, 16, 34, 46, 40, 22, 7, 1; ... MAPLE T:= proc(n, k) option remember; if k=0 then 1 else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2)) fi; end: seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019 MATHEMATICA T[n_, k_]:= T[n, k]= If[k==0, 1, Sum[Binomial[n-1-2*j, k-1], {j, 0, Floor[(n-1)/2]}]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *) PROG (PARI) T(n, k) = if(k==0, 1, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ G. C. Greubel, Nov 20 2019 (Magma) function T(n, k) if k eq 0 then return 1; else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]); end if; return T; end function; [T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019 (Sage) @CachedFunction def T(n, k): if (k==0): 1 else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2))) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019 CROSSREFS Cf. A005578, A007318, A103451. Sequence in context: A066287 A059260 A239473 * A257543 A081372 A101489 Adjacent sequences: A135226 A135227 A135228 * A135230 A135231 A135232 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Nov 23 2007 EXTENSIONS Offset changed by G. C. Greubel, Nov 20 2019 STATUS approved

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Last modified August 7 05:36 EDT 2024. Contains 375008 sequences. (Running on oeis4.)