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A154283
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Irregular triangle read by rows: T(n,k) = (-1) times coefficient of x^k in (x-1)^(2*n+1) * sum_{k>=0} (k*(k+1)/2)^n *x^(k-1), 0<=k<=2*n-2.
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8
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1, 1, 4, 1, 1, 20, 48, 20, 1, 1, 72, 603, 1168, 603, 72, 1, 1, 232, 5158, 27664, 47290, 27664, 5158, 232, 1, 1, 716, 37257, 450048, 1822014, 2864328, 1822014, 450048, 37257, 716, 1, 1, 2172, 247236, 6030140, 49258935, 163809288, 242384856, 163809288, 49258935
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OFFSET
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1,3
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COMMENTS
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Row sums are in A000680.
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LINKS
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Table of n, a(n) for n=1..45.
H. Prodinger, On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs , arXiv:1102.5186 [math.CO], 2011.
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FORMULA
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From Yahia Kahloune, Jan 29 2014: (Start)
Using these coefficients we can obtain formulas for the sums Sum_{i=1..n} binomial(1+i,2)^p. Let us define a(k,e,p) = Sum_{i=0..k-2} (-1)^i*binomial(2*p+1,i)*binomial(k-i,2)^p.
For example, a(2,2,p) = 1; a(3,2,p) = 3^p - (2*p+1); a(4,2,2) = 6^p - (2*p+1)*3^p + binomial(2*p+1,2); a(5,2,p) = 10^p - (2*p+1)*6^p + binomial(2*p+1,2)*3^p - binomial(2*p+1,3).
Then we have the formula : Sum_{i=1..n} binomial(1+i,2)^p = Sum_{i=0..2*p-2} a(2+i,2,p)*binomial(n+2+i,2*p+1).
Example: Sum_{i=1..8} binomial(1+i,2)^4 = binomial(10,9) +72* binomial(11,9) + 603*binomial(12,9) + 1168*binomial(13,9) + 603*binomial(14,9) + 72*binomial(15,9) + binomial(16,9) = 2550756. (End)
From Peter Bala, Dec 21 2019; (Start)
E.g.f. as a continued fraction: (1-x)/(1-x + ( 1-exp((1-x)^2*t))*x/(1-x + (1-exp(2*(1-x)^2*t))*x/(1-x + (1-exp(3*(1-x)^2*t))*x/(1-x + ... )))) = 1 + x*t + x*(x^2 + 4*x + 1)*t^2/2! + x*(x^4 + 20*x^3 + 48*x^2 + 20*x + 1)*t^3/3! + ... (use Prodinger equation 1.1).
The sequence of alternating row sums (unsigned) [1, 1, 2, 10, 104, 1816,...] appears to be A005799. (End)
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EXAMPLE
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Triangle begins:
1;
1, 4, 1;
1, 20, 48, 20, 1;
1, 72, 603, 1168, 603, 72, 1;
1, 232, 5158, 27664, 47290, 27664, 5158, 232, 1;
1, 716, 37257, 450048, 1822014, 2864328, 1822014, 450048, 37257, 716, 1;
1, 2172, 247236, 6030140, 49258935, 163809288, 242384856, 163809288, 49258935, 6030140, 247236, 2172, 1;
1, 6544, 1568215, 72338144, 1086859301, 6727188848, 19323413187, 27306899520, 19323413187, 6727188848, 1086859301, 72338144, 1568215, 6544, 1;
1, 19664, 9703890, 811888600, 21147576440, 225167210712, 1130781824398, 2898916824320, 3950966047950, 2898916824320, 1130781824398, 225167210712, 21147576440, 811888600, 9703890, 19664, 1;
...
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MAPLE
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A154283 := proc(n, k)
(1-x)^(2*n+1)*add( (l*(l+1)/2)^n*x^(l-1), l=0..k+1) ;
coeftayl(%, x=0, k) ;
end proc: # R. J. Mathar, Feb 01 2013
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MATHEMATICA
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Clear[p, x, n]; p[x_, n_] = (1-x)^(2*n + 1)*Sum[(k*(k + 1)/2)^n*x^k, {k, 0, Infinity}]/x;
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]
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CROSSREFS
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Cf. A000680, A005799.
Sequence in context: A176422 A156586 A181544 * A185946 A015113 A016519
Adjacent sequences: A154280 A154281 A154282 * A154284 A154285 A154286
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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Roger L. Bagula, Jan 06 2009
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EXTENSIONS
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Edited by N. J. A. Sloane, Jan 30 2014 following suggestions from Yahia Kahloune (among other things, the signs of all terms have been reversed).
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STATUS
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approved
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