cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A067588 Total number of parts in all partitions of n into odd parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 14, 19, 26, 36, 48, 62, 82, 104, 132, 169, 210, 260, 324, 396, 484, 592, 714, 860, 1036, 1238, 1474, 1756, 2078, 2452, 2894, 3396, 3976, 4654, 5422, 6309, 7332, 8490, 9816, 11338, 13060, 15018, 17254, 19774, 22630, 25878, 29524, 33642
Offset: 0

Views

Author

Naohiro Nomoto, Jan 31 2002

Keywords

Comments

Starting with "1" = triangle A097304 * [1, 2, 3, ...]. - Gary W. Adamson, Apr 09 2010

Links

Crossrefs

Cf. A067589, A006128, A066897, A066898.
Cf. A097304. - Gary W. Adamson, Apr 09 2010

Formula

G.f.: G(x)*H(x) where G(x) = Sum_{k>=1} x^(2*k-1)/(1-x^(2*k-1)) is g.f. for the number of odd divisors of n (cf. A001227) and H(x) = Product_{k>=1} (1+x^k) is g.f. for the number of partitions of n into odd parts (cf. A000009). Convolution of A001227 and A000009: Sum_{k=0..n} A001227(k)*A000009(n-k). - Vladeta Jovovic, Feb 04 2002
G.f.: Sum_{n>0} n*x^n/Product_{k=1..n} (1-x^(2*k)). - Vladeta Jovovic, Dec 15 2003
a(n) ~ 3^(1/4) * (2*gamma + log(48*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (8*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

Extensions

Corrected by James Sellers, May 31 2007

A152140 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n into k odd parts.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 3, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1
Offset: 0

Views

Author

R. J. Mathar, Sep 25 2009, offset corrected Jul 09 2012

Keywords

Comments

The number of partitions of n into k odd parts is equal to the number of partitions of (n+k)/2 into k parts; or equivalently the number of partitions of (n-k)/2 into at most k parts. - Franklin T. Adams-Watters, Sep 25 2009

Examples

			n= 0, k= 0: [];
n= 1, k= 1: [1] ;
n= 2, k= 2: [1, 1] ;
n= 3, k= 1: [3] ;
n= 3, k= 3: [1, 1, 1] ;
n= 4, k= 2: [1, 3] ;
n= 4, k= 4: [1, 1, 1, 1];
n= 5, k= 1: [5];
n= 5, k= 3: [1, 1, 3];
n= 5, k= 5: [1, 1, 1, 1, 1];
n= 6, k= 2: [3, 3] or [1, 5];
n= 6, k= 4: [1, 1, 1, 3];
n= 6, k= 6: [1, 1, 1, 1, 1, 1];
Triangle begins:
1
0 1
0 0 1
0 1 0 1
0 0 1 0 1
0 1 0 1 0 1
0 0 2 0 1 0 1
0 1 0 2 0 1 0 1
0 0 2 0 2 0 1 0 1
0 1 0 3 0 2 0 1 0 1
0 0 3 0 3 0 2 0 1 0 1
0 1 0 4 0 3 0 2 0 1 0 1
0 0 3 0 5 0 3 0 2 0 1 0 1
0 1 0 5 0 5 0 3 0 2 0 1 0 1
0 0 4 0 6 0 5 0 3 0 2 0 1 0 1
0 1 0 7 0 7 0 5 0 3 0 2 0 1 0 1
0 0 4 0 9 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 8 0 10 0 7 0 5 0 3 0 2 0 1 0 1
0 0 5 0 11 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 10 0 13 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 5 0 15 0 14 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 12 0 18 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 6 0 18 0 20 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 14 0 23 0 21 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 6 0 23 0 26 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 16 0 30 0 28 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 7 0 27 0 35 0 29 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 19 0 37 0 38 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 7 0 34 0 44 0 40 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 21 0 47 0 49 0 41 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 8 0 39 0 58 0 52 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 24 0 57 0 65 0 54 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 8 0 47 0 71 0 70 0 55 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 27 0 70 0 82 0 73 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 0 9 0 54 0 90 0 89 0 75 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
0 1 0 30 0 84 0 105 0 94 0 76 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1

Links

Crossrefs

Cf. A000009 (row sums), A097304, A107379, A152146, A152157.

Programs

  • Maple
    b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 elif irem(i, 2)=0 then b(n, i-1)
      else []; for j from 0 to n/i do zip((x, y)->x+y, %,
      [0$j, b(n-i*j, i-2)], 0) od; %[] fi
    end:
T:= n-> b(n$2):
seq(T(n), n=0..13);  # Alois P. Heinz, May 31 2013
  • Mathematica
    nn = 10; CoefficientList[
Series[Product[1/(1 - y x^i), {i, 1, nn, 2}], {x, 0, nn}], {x, y}] (* Geoffrey Critzer, May 31 2013 *)

A097301 Numerators of rationals used in the Euler-Maclaurin type derivation of Stirling's formula for N!.

Original entry on oeis.org

1, -1, 2, -3, 3360, -995040, 39916800, -656924748480, 1214047650816000, -169382556838010880, 15749593891765493760000, -4054844479616799289344000, 34017686450062663131463680000, -11402327189708082115897599590400000, 189528830020089532044244068728832000000
Offset: 0

Views

Author

Wolfdieter Lang, Aug 13 2004

Keywords

Comments

Denominators are given in A097302.
The e.g.f. sum( A(2*n+1)*(x^(2*n+1))/(2*n+1)!,n=0..infinity) appears in the Stirling-formula derivation for N! with x=1/N in the exponent and the formula for A(2*n+1):=a(n)/A097302(n), n>=0, is given below. For Stirling's formula see A001163 and A001164.
The rationals A(2*n+1) = B(n):= (2*n)!*Bernoulli(2*(n+1))/(2*(n+1)) = a(n)/A097304(n) with A(2*n):=0 are the logarithmic transform of the rational sequence {A001163(n)/A001164(n)} (inverse of the sequence transform EXP)

References

  • Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, p. 87.

Links

Formula

a(n)=numerator(B(n)) with B(n):=Bernoulli(2*n+2)*(2*n)!/(2*n+2) and Bernoulli(n)= A027641(n)/A027642(n).
Showing 1-3 of 3 results.

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