A186445 -id:A186445 - cp's OEIS frontend

A139039 A triangular central symmetric sequence based on the sequence A003269: if m <= floor(n/2), t(n,m) = A003269(m+2), otherwise t(n,m) = A003269(n - (m+2)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, May 31 2008

Row sums: {1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 20, ...}. [Is this A186445 or A080078? - N. J. A. Sloane, Feb 10 2013]

The A003269 sequence is pushed back twice, so that the triangle is not almost all ones.

			{1},
{1, 1},
{1, 1, 1},
{1, 1, 1, 1},
{1, 1, 1, 1, 1},
{1, 1, 1, 1, 1, 1},
{1, 1, 1, 2, 1, 1, 1},
{1, 1, 1, 2, 2, 1, 1, 1},
{1, 1, 1, 2, 3, 2, 1, 1, 1},
{1, 1, 1, 2, 3, 3, 2, 1, 1, 1},
{1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1}

Cf. A139147, A003269.

Clear[a]; a[ -2] = 0; a[ -1] = 1; a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 4]; (* A003269 *) Table[If[m <= Floor[n/2],a[m],a[n-m] ] ,{n,0,10},{m,0,n}]

a(n) = a(n-1) + a(n-4); t(n,m) = a(m) if m <= floor(n/2), a(n-m) otherwise.

Non-ASCII characters removed and Mathematica code corrected by Wouter Meeussen, Feb 10 2013

A213424 Number of partitions of n in which all parts are >= 2 and the largest part occurs at least five times.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 4, 6, 6, 8, 8, 12, 10, 15, 15, 19, 21, 26, 26, 34, 36, 45, 47, 59, 61, 76, 83, 97, 107, 128, 137, 165, 179, 210, 231, 271, 296, 345, 380, 438, 485, 561, 614, 708, 783, 893, 991, 1129, 1246, 1420, 1572, 1781

Offset: 10

Mircea Merca, Jun 11 2012

nonn

			For n = 20 we have three partitions: {[4+4+4+4+4], [3+3+3+3+3+3+2], [2+2+2+2+2+2+2+2+2+2]}, so a(20) = 3.

Cf. A000041, A186445, A213423.

nmax = 100; Drop[CoefficientList[Series[(1 - x)^2*(1 - x^2)*(1 - x^3)*(1 - x^4) / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x], 10] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = A186445(n) - 2*A186445(n-1) + A186445(n-2).
G.f.: (1-x)*Product_{k>4} 1/(1-x^k).
a(n) ~ Pi^5 * exp(Pi*sqrt(2*n/3))/ (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jul 05 2025

cp's OEIS Frontend

A139039 A triangular central symmetric sequence based on the sequence A003269: if m <= floor(n/2), t(n,m) = A003269(m+2), otherwise t(n,m) = A003269(n - (m+2)).

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