A128506 -id:A128506 - cp's OEIS frontend

A128507 Denominators of partial sums for a series for 3sqrt(2)(Pi^3)/2^7.

1, 27, 3375, 1157625, 31255875, 41601569625, 91398648466125, 91398648466125, 449041559914072125, 3079976059450620705375, 439996579921517243625, 5353438387905100303185375, 669179798488137537898171875

Offset: 0

Wolfdieter Lang, Apr 04 2007

The numerators are given in A128506.

See the comments and the W. Lang link under A128506.

			Rationals r(n): [1, 28/27, 3473/3375, 1187864/1157625, 32115203/31255875,...].
3*sqrt(2)*(Pi^3)/2^7 = +1/1^3 +1/3^3 -1/5^3 -1/7^3 +1/9^3 +1/11^3 -1/13^3 -1/15^3 ++--

G. C. Greubel, Table of n, a(n) for n = 0..385

r[n_]:= Sum[(1/2)*((1-I)*I^k+(1+I)*(-I)^k)/(2k+1)^3, {k,0,n}]; Denominator[Table[r[n], {n,0,30}]] (* G. C. Greubel, Mar 28 2018 *)

{r(n) = sum(k=0,n, ((1-I)*I^k + (1+I)*(-I)^k)/(2*(2*k+1)^3))};
for(n=0,30, print1(denominator(r(n)), ", ")) \\ G. C. Greubel, Mar 28 2018

a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} S(2*k,sqrt(2))/(2*k+1)^3 with Chebyshev's S-Polynomials S(2*k,sqrt(2))=[1,1,-1,-1] periodic sequence with period 4. See A057077.

A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)).

Original entry on oeis.org

1, 4, 10, -4, 20, -20, 35, -60, 10, 56, -140, 60, 84, -280, 210, -20, 120, -504, 560, -140, 165, -840, 1260, -560, 35, 220, -1320, 2520, -1680, 280, 286, -1980, 4620, -4200, 1260, -56, 364, -2860, 7920, -9240, 4200, -504, 455, -4004, 12870, -18480, 11550, -2520, 84, 560, -5460, 20020, -34320

Offset: 0

Wolfdieter Lang, Apr 04 2007

S3(n,x) := Sum_{k=0..n} S(n-k,x)*S2(k,x) = Sum_{m=0..floor(n/2)} a(n,m)*x^(n-2*m) with the second convolution S2(n,x) given by array A128503.

Row polynomials P3(n,x) := Sum_{m=0..floor(n/2)} a(n,m)*x^m (increasing powers of x).

			  1;
  4;
  10,   -4;
  20,  -20;
  35,  -60,  10;
  56, -140,  60;
  84, -280, 210,  -20;
  120,-504, 560, -140;
  ...
n=4: [35,-60,10] stands also for the row polynomial P3(4,x) = 35-60*x+10*x^2.

Wolfdieter Lang, First 15 rows and more.

Row sums (signed array) give A128506. Unsigned row sums are A001872.

Cf. A128503 (k=2 convolution).

a(n,m) = binomial(n-m+3,3)*binomial(n-m,m)*(-1)^m, m = 0..floor(n/2), n >= 0.
a(n,m) = binomial(m+3,3)*binomial(n-m+3,m+3)*(-1)^m, m = 0..floor(n/2), n >= 0.
G.f. for S3(n,x): 1/(1-x*z+z^2)^4.
G.f. for P3(n,x): 1/(1-z+x*z^2)^4.

Name edited by Petros Hadjicostas, Sep 04 2019

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A128507 Denominators of partial sums for a series for 3sqrt(2)(Pi^3)/2^7.

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A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)).

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cp's OEIS Frontend

A128507 Denominators of partial sums for a series for 3*sqrt(2)*(Pi^3)/2^7.

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Keywords

Comments

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Mathematica

PARI

Formula

A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)).

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A128507 Denominators of partial sums for a series for 3sqrt(2)(Pi^3)/2^7.