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A054143
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Triangular array T given by T(n,k)=Sum{C(i,j): 0<=j<=i-n+k, n-k<=i<=n}.
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8
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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. [From Clark Kimberling, Aug 7 2011]
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FORMULA
| T(n,k)=Sum{C(i,j): 0<=j<=i-n+k, n-k<=i<=n}.
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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
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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
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MATHEMATICA
| 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 *)
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CROSSREFS
| 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).
Diagonal sums give A005672. - Paul Barry (pbarry(AT)wit.ie), Feb 07 2003
Sequence in context: A028861 A081521 A086273 * A104746 A193969 A169838
Adjacent sequences: A054140 A054141 A054142 * A054144 A054145 A054146
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Mar 18 2000
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