A182712 -id:A182712 - cp's OEIS frontend

A182716 Number of 2's in all partitions of 2n that do not contain 1 as a part.

0, 1, 2, 4, 8, 15, 27, 48, 82, 137, 225, 362, 572, 892, 1370, 2078, 3117, 4624, 6791, 9885, 14263, 20416, 29007, 40921, 57345, 79864, 110565, 152211, 208435, 283982, 385048, 519695, 698346, 934477, 1245439, 1653485, 2187108, 2882686, 3786497, 4957324, 6469625

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas, and Gideon Vos, Large-N correlation functions in N = 2 superconformal QCD, arXiv preprint arXiv:1610.07612 [hep-th], 2016.

Cf. A182742. Bisection of A182712.

b:= proc(n,i) option remember; local r;
      if n<=0 or i<2 then 0
    elif i=2 then `if`(irem(n,2,'r')=0,r,0)
    else b(n,i-1) +b(n-i,i)
      fi
    end:
a:= n-> b(2*n,2*n):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 03 2010

b[n_, i_] := b[n, i] = Module[{q, r}, Which[n <= 0 || i<2, 0, i==2, {q, r} = QuotientRemainder[n, 2]; If[r==0, q, 0], True, b[n, i-1]+b[n-i, i]]]; a[n_] := b[2n, 2n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)

More terms from Alois P. Heinz, Dec 03 2010

A182725 Bisection of A005291.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 28, 51, 92, 163, 285, 490, 833, 1396, 2313, 3789

Offset: 1

Omar E. Pol, Dec 02 2010

This sequence contains the string "1, 2, 4, 8, 15, 28, 51, 92" the same as A005682 and A029907. The next terms is similar. I think that A005291 is a sequence formed from two similar mechanisms but not from only one.

Cf. A005291, A182712, A182722, A182726.

A182726 Bisection of A005291.

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 30, 58, 108, 196, 348, 605, 1034, 1740, 2887, 4730

Offset: 1

Omar E. Pol, Dec 02 2010

Cf. A005291, A182712, A182722, A182725.

A195308 a(n) = A005291(n) + A005291(n+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 58, 81, 109, 150, 200, 271, 359, 481, 633, 838, 1095, 1438, 1867, 2430, 3136, 4053, 5200, 6676, 8519, 10871, 13802, 17514, 22129, 27940, 35141, 44155, 55299, 69179, 86286, 107495, 133562, 165744, 205188, 253691, 312975, 385619

Offset: 1

Omar E. Pol, Feb 03 2012

nonn

This sequence arises from A005291 in the same way as A000041 arises from A182712.

Observation: a(3)..a(13) coincide with a sequence related to Stirling's numbers from the Jordan's book.

Charles Jordan, Calculus of finite differences, Chelsea Publishing Co., 1965, chapter IV, pp. 153-155.

Cf. A000041, A005291, A087787, A100818, A182712, A194439, A195017.

a(n) = A000041(n-2), 2 <= n <= 11. - Omar E. Pol, Feb 24 2013

More terms from Amiram Eldar, May 17 2025

cp's OEIS Frontend

A182716 Number of 2's in all partitions of 2n that do not contain 1 as a part.

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A182725 Bisection of A005291.

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A182726 Bisection of A005291.

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A195308 a(n) = A005291(n) + A005291(n+1).

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