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 A054143 Triangular array T given by T(n,k) = Sum_{0<=j<=i-n+k, n-k<=i<=n} C(i,j). 11
 1, 1, 3, 1, 4, 7, 1, 5, 11, 15, 1, 6, 16, 26, 31, 1, 7, 22, 42, 57, 63, 1, 8, 29, 64, 99, 120, 127, 1, 9, 37, 93, 163, 219, 247, 255, 1, 10, 46, 130, 256, 382, 466, 502, 511, 1, 11, 56, 176, 386, 638, 848, 968, 1013, 1023, 1, 12, 67, 232, 562, 1024, 1486, 1816, 1981, 2036, 2047 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums given by A001787. T(n, n) = -1 + 2^(n+1). T(2n, n) = 4^n. T(2n+1, n) = A000346(n). T(2n-1, n) = A032443(n). A054143 is the fission of the polynomial sequence ((x+1^n) by the polynomial sequence (q(n,x)) given by q(n,x) = x^n + x^(n-1) + ... + x + 1. See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011 LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened FORMULA T(n,k) = Sum_{0<=j<=i-n+k, n-k<=i<=n} binomial(i,j). T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k-1) - 2*T(n-2,k-2), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 30 2013 EXAMPLE First six rows:   1;   1,  3;   1,  4,  7;   1,  5, 11, 15;   1,  6, 16, 26, 31;   1,  7, 22, 42, 57, 63; MAPLE A054143_row := proc(n) add(add(binomial(n, n-i)*x^(k+1), i=0..k), k=0..n-1); coeffs(sort(%)) end; seq(print(A054143_row(n)), n=1..6); # Peter Luschny, Sep 29 2011 MATHEMATICA (* First program *) z=10; p[n_, x_]:=(x+1)^n; q[0, x_]:=1; q[n_, x_]:=x*q[n-1, x]+1; p1[n_, k_]:=Coefficient[p[n, x], x^k]; p1[n_, 0]:=p[n, x]/.x->0; d[n_, x_]:=Sum[p1[n, k]*q[n-1-k, x], {k, 0, n-1}] h[n_]:=CoefficientList[d[n, x], {x}] TableForm[Table[Reverse[h[n]], {n, 0, z}]] Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A054143 *) TableForm[Table[h[n], {n, 0, z}]] Flatten[Table[h[n], {n, -1, z}]] (* A104709 *) (* Second program *) Table[Sum[Binomial[i, j], {i, n-k, n}, {j, 0, i-n+k}], {n, 0, 12}, {k, 0, n}]// Flatten (* G. C. Greubel, Aug 01 2019 *) PROG (PARI) T(n, k) = sum(i=n-k, n, sum(j=0, i-n+k, binomial(i, j))); for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 01 2019 (MAGMA) T:= func< n, k | (&+[ (&+[ Binomial(i, j): j in [0..i-n+k]]): i in [n-k..n]]) >; [T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019 (Sage) def T(n, k): return sum(sum( binomial(i, j) for j in (0..i-n+k)) for i in (n-k..n)) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019 (GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([n-k..n], i-> Sum([0..i-n+k], j-> Binomial(i, j) ))))); # G. C. Greubel, Aug 01 2019 CROSSREFS Cf. A000346, A001787, A032443. Diagonal sums give A005672. - Paul Barry, Feb 07 2003 Sequence in context: A213224 A210218 A086273 * A104746 A208339 A328463 Adjacent sequences:  A054140 A054141 A054142 * A054144 A054145 A054146 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 18 2000 STATUS approved

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Last modified December 16 06:18 EST 2019. Contains 330016 sequences. (Running on oeis4.)