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A154283 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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums are in A000680.

LINKS

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.

FORMULA

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)

EXAMPLE

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;

...

MAPLE

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

MATHEMATICA

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[%]

CROSSREFS

Cf. A000680, A005799.

Sequence in context: A176422 A156586 A181544 * A185946 A015113 A016519

Adjacent sequences:  A154280 A154281 A154282 * A154284 A154285 A154286

KEYWORD

nonn,easy,tabf

AUTHOR

Roger L. Bagula, Jan 06 2009

EXTENSIONS

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).

STATUS

approved

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Last modified January 17 20:36 EST 2020. Contains 330987 sequences. (Running on oeis4.)