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A201454 Triangle of denominators of dual coefficients of Faulhaber. 1
1, 3, 3, 5, 3, 15, 7, 5, 3, 105, 9, 21, 15, 9, 105, 11, 9, 21, 3, 9, 231, 13, 11, 3, 7, 5, 3, 15015, 15, 39, 165, 9, 15, 5, 45, 2145, 17, 5, 13, 55, 9, 15, 15, 45, 36465, 19, 17, 5, 13, 55, 3, 35, 1, 5, 969969, 21, 57, 17, 21, 13, 33, 63, 7, 5, 63, 4849845 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Sum((k*(k + 1))^(m), k=0..N-1)=Sum(F(m,i)*N^(2*m-2*i+1),i=0..m), m=0,1,2,...

The coefficients F(m,i) are dual to Faulhaber coefficients, because they are obtained from the inverse expression Sum((k*(k + 1))^(m), k=0..N-1) to Faulhaber's formula from Sum((k)^(2*m-1), k=0..N-1) and there holds the identity F(m+i-1,i)=(-1)^i Fe(-m,i), where Fe(-m,i)=A093558(-m,i)/A093559(-m,i) is a Faulhaber coefficient for the sums of even powers of the first N-1 integers (for details see the link, from p. 19).

LINKS

Table of n, a(n) for n=0..65.

Askar Dzhumadil'daev, Damir Yeliussizov, Power sums of binomial coefficients, Journal of Integer Sequences, 16 (2013), Article 13.1.6

FORMULA

a(m,k) = denominator(F(m,k)) with F(m,k) = (1/(2*m-2*k+1)) * sum(i=0..2*k, binomial(m,2*k-i) * binomial(2*m-2*k+i,i) * Bernoulli(i) ).

A recursion is given by F(x,0) = 1/(2*x+1) and 2*(m-k)*(2*m-2*k+1)*F(m,k)=2*m*(2*m-1)*F(m-1,k)+m*(m-1)*F(m-2,k-1).

EXAMPLE

Triangle begins:

1;

3,  3;

5,  3,  15;

7,  5,  3,   10;

9,  21, 15,  9,  105;

11, 9,  21,  3,  9,   231;

13, 11, 3,   7,  5,   3,   15015;

15, 39, 165, 9,  15,  5,   45,    2145;

17, 5,  13,  55, 9,   15,  15,    45,   36465;

19, 17, 5,   13, 55,  3,   35,    1,    5,    969969;

21, 57, 17,  21, 13,  33,  63,    7,    5,    63,    4849845;

etc.

MATHEMATICA

f[m_, k_] := (1/(2*m - 2*k + 1))* Sum[Binomial[m, 2*k - i]*Binomial[2*m - 2*k + i, i]*BernoulliB[i], {i, 0, 2 k}];

a[m_, k_] := f[m, k] // Denominator;

Table[a[m, k], {m, 0, 10}, {k, 0, m}] // Flatten (* Jean-Fran├žois Alcover, Jan 18 2013 *)

PROG

(MAGMA) [Denominator((1/(2*m-2*k+1))*&+[Binomial(m, 2*k-i)*Binomial(2*m-2*k+i, i)*BernoulliNumber(i): i in [0..2*k]]): k in [0..m], m in [0..10]]; // Bruno Berselli, Jan 21 2013

CROSSREFS

Cf. A201453.

Sequence in context: A298398 A094439 A122037 * A008316 A290284 A258802

Adjacent sequences:  A201451 A201452 A201453 * A201455 A201456 A201457

KEYWORD

nonn,frac,tabl,easy

AUTHOR

Damir Yeliussizov, Jan 09 2013

STATUS

approved

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Last modified August 22 05:00 EDT 2019. Contains 326172 sequences. (Running on oeis4.)