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A176665
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Triangle of polynomial coefficients of p(x,n) = Sum_{k=0..n} (k + 1)^n * k! * binomial(x, k), read by rows.
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1
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1, 1, 2, 1, -5, 9, 1, 109, -165, 64, 1, -3303, 6188, -3494, 625, 1, 169711, -357254, 254434, -74635, 7776, 1, -13084359, 30063342, -24927719, 9549230, -1718079, 117649, 1, 1417404703, -3486909736, 3229823067, -1474126800, 354928391, -43216649, 2097152
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OFFSET
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0,3
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COMMENTS
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Row sums are: A083318 = {1, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, ...}.
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LINKS
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FORMULA
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Let p(x,n) = Sum_{k=0..n} (k + 1)^n * k! * binomial(x, k) then the number triangle is given by T(n, m) = coefficients( p(x,n) ).
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EXAMPLE
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Triangle begins as:
1;
1, 2;
1, -5, 9;
1, 109, -165, 64;
1, -3303, 6188, -3494, 625;
1, 169711, -357254, 254434, -74635, 7776;
1, -13084359, 30063342, -24927719, 9549230, -1718079, 117649;
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MATHEMATICA
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(* First program *)
p[x_, n_]:= Sum[(k+1)^n*k!*Binomial[x, k], {k, 0, n}];
Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]//Flatten
(* Second program *)
f[n_]:= CoefficientList[Sum[(k+1)^n*Product[x-j, {j, 0, k-1}], {k, 0, n}], x];
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PROG
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(Sage)
def p(n, x): return sum( (k+1)^n*factorial(k)*binomial(x, k) for k in (0..n))
flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 07 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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