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A259477
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Triangle of numbers where T(n,k) is the number of k-dimensional faces on a partially truncated n-dimensional simplex, 0 <= k <= n.
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2
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1, 2, 1, 6, 6, 1, 12, 18, 8, 1, 20, 40, 30, 10, 1, 30, 75, 80, 45, 12, 1, 42, 126, 175, 140, 63, 14, 1, 56, 196, 336, 350, 224, 84, 16, 1, 72, 288, 588, 756, 630, 336, 108, 18, 1, 90, 405, 960, 1470, 1512, 1050, 480, 135, 20, 1, 110, 550, 1485, 2640, 3234, 2772, 1650, 660, 165, 22, 1
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OFFSET
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0,2
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LINKS
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FORMULA
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T(n,0) = n*(n+1), n > 0; T(n,k) = (n+1-k)*binomial(n+1,k+1), 1 <= k <= n.
E.g.f.: ((x+1)*(z+1)+1)*exp(z)*(exp(x*z)-1)/x + 1.
O.g.f.: (1/(1-(x+1)*y)^2-(x+1)/(1-y)^2)/x + 1/((1-(x+1)y)*(1-y))+1.
G.f. for rows (n > 0): (((x+1)^n-1)*(x+n+2))/x-n. (End)
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EXAMPLE
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Triangle begins:
1;
2, 1;
6, 6, 1;
12, 18, 8, 1;
20, 40, 30, 10, 1;
...
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MATHEMATICA
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Join @@ (CoefficientList[#,
x] & /@ (Expand[
D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z] - 1)/x + 1, {z, #}] /.
Flatten[Table[
CoefficientList[
D[(1/(1 - (x + 1)*y)^2 - (x + 1)/(1 - y)^2)/x +
1/((1 - (x + 1) y)*(1 - y)) + 1, {y, k}]/Factorial[k] /. y -> 0,
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PROG
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(PARI) T(n, k)=max(if(k, n+1-k, n)*binomial(n+1, k+1), 1)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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