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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A061550 a(n) = (2*n+1)*(2*n+3)*(2*n+5).

Original entry on oeis.org

15, 105, 315, 693, 1287, 2145, 3315, 4845, 6783, 9177, 12075, 15525, 19575, 24273, 29667, 35805, 42735, 50505, 59163, 68757, 79335, 90945, 103635, 117453, 132447, 148665, 166155, 184965, 205143, 226737, 249795, 274365, 300495, 328233, 357627
Offset: 0

Views

Author

Jason Earls, Jun 12 2001

Keywords

Comments

sum(1/a(k), k=0..n) = 1/12 - 1/((8*n+12)*(2*n+5)). Jolley equation 209 (offset adjusted). - Gary Detlefs, Sep 20 2011

References

  • L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 40

Links

Crossrefs

Cf. A005408.

Programs

  • Maple
    For n from 0 to 100 do (2*n+1)*(2*n+3)*(2*n+5) end do;
  • Mathematica
    f[n_] := n/GCD[n, 4]; Table[ f[n] f[n + 2] f[n + 4], {n, 1, 70, 2}] (* Robert G. Wilson v, Jan 14 2011 *)
Times@@@(#+{1,3,5}&)/@(2Range[0,35]) (* Harvey P. Dale, Feb 13 2011 *)
Table[(2*n + 1)*(2*n + 3)*(2*n + 5), {n,35}] (* T. D. Noe, Feb 13 2011 *)
  • PARI
    a(n) = { (2*n + 1)*(2*n + 3)*(2*n + 5) } \\ Harry J. Smith, Jul 24 2009

Formula

a(n) = A162540(n)/3.
1/15 + 1/105 + 1/315...= 1/12 [Jolley, eq. 209]
sum_{i=0..n-1} a(i) = A196506(n), partial sums [Jolley eq (43)]. - R. J. Mathar, Mar 24 2011
sum_{i=0..infinity} (-1)^i/a(i) = Pi/8-1/3 = 0.0593657... [Jolley eq 240]
a(n)=(-1)^(n+1)*(4*n^2+12*n+7)/Integral_{x=0..Pi/2} (cos((2*n+3)*x))*(sin(x))^2 dx. - Francesco Daddi, Aug 03 2011
G.f. ( 15+45*x-15*x^2+3*x^3 ) / (x-1)^4. - R. J. Mathar, Oct 03 2011

Extensions

Better description and more terms from Larry Reeves (larryr(AT)acm.org), Jun 19 2001

A381059 Array read by ascending antidiagonals: A(n,k) = numerator(binomial(n-1/2,k)) with k >=0.

Original entry on oeis.org

1, 1, -1, 1, 1, 3, 1, 3, -1, -5, 1, 5, 3, 1, 35, 1, 7, 15, -1, -5, -63, 1, 9, 35, 5, 3, 7, 231, 1, 11, 63, 35, -5, -3, -21, -429, 1, 13, 99, 105, 35, 3, 7, 33, 6435, 1, 15, 143, 231, 315, -7, -5, -9, -429, -12155, 1, 17, 195, 429, 1155, 63, 7, 5, 99, 715, 46189
Offset: 0

Views

Author

Stefano Spezia, Feb 12 2025

Keywords

Comments

Numerators of the binomial coefficients for half-integers. The denominators are given by the absolute values of A173755.

Examples

			The array of the binomial coefficients for half-integers begins as:
  1, -1/2,  3/8,  -5/16,   35/128, -63/256, ...
  1,  1/2, -1/8,   1/16,   -5/128,   7/256, ...
  1,  3/2,  3/8,  -1/16,    3/128,  -3/256, ...
  1,  5/2, 15/8,   5/16,   -5/128,   3/256, ...
  1,  7/2, 35/8,  35/16,   35/128,  -7/256, ...
  1,  9/2, 63/8, 105/16,  315/128,  63/256, ...
  1, 11/2, 99/8, 231/16, 1155/128, 693/256, ...
  ...

Links

Crossrefs

Cf. A000466, A161200, A161202, A162540, A173755.
Columns k=0..1 give A000012, A060747.
Row n=1 gives A002596.
Main diagonal gives A001790.

Programs

  • Mathematica
    A[n_,k_]:=Numerator[Binomial[n-1/2,k]]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=Numerator[(2n-1)!!/((2(n-k)-1)!!2^k k!)]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

A(n,k) = numerator((2*n - 1)!!/((2*(n - k) - 1)!!*2^k*k!)).
A(n,2) = A000466(n-1) for n > 0.
A(n,3) = A162540(n-3) for n > 3.
A(0,n) = (-1)^n*A001790(n).
abs(A(2,n)) = abs(A161200(n)).
abs(A(3,n)) = abs(A161202(n)).
Showing 1-2 of 2 results.

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