A000098 Number of partitions of n if there are two kinds of 1, two kinds of 2 and two kinds of 3.

1, 2, 5, 10, 19, 33, 57, 92, 147, 227, 345, 512, 752, 1083, 1545, 2174, 3031, 4179, 5719, 7752, 10438, 13946, 18519, 24428, 32051, 41805, 54265, 70079, 90102, 115318, 147005, 186626, 236064, 297492, 373645, 467707

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also number of partitions of 2*n+1 with exactly 3 odd parts (offset 1). - Vladeta Jovovic, Jan 12 2005

Convolution of A000041 and A001399. - Vaclav Kotesovec, Aug 18 2015

Also the sum of binomial(D(p),3) over partitions p of n+6, where D(p) is the number of different sizes of parts in p. - Emily Anible, May 13 2018

Examples

			a(3)=10 because we have 3, 3', 2+1, 2+1', 2'+1, 2'+1', 1+1+1, 1+1+1', 1+1'+1' and 1'+1'+1'.
		

References

• H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Álvaro Gutiérrez and Mercedes H. Rosas, Partial symmetries of iterated plethysms, arXiv:2201.00240 [math.CO], 2022.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000070, A008951, A000097, A000710.

Cf. A000041, A001399.

Fourth column of Riordan triangle A008951 and of triangle A103923.

Programs

Mathematica

CoefficientList[1/((1-x)*(1-x^2)*(1-x^3)*QPochhammer[x]) + O[x]^40, x] (* Jean-François Alcover, Feb 04 2016 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@3], {n,0,35}] (* Robert Price, Jul 28 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[, ] = 0;
a[n_] := T[n + 6, 3];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Formula

Euler transform of 2 2 2 1 1 1 1...

G.f.: 1/((1-x)(1-x^2)(1-x^3)*Product_{k>=1} (1-x^k)).

a(n) = Sum_{j=0..floor(n/3)} A000097(n-3*j), n >= 0.

a(n) ~ sqrt(n) * exp(Pi*sqrt(2*n/3)) / (2*sqrt(2)*Pi^3). - Vaclav Kotesovec, Aug 18 2015

Extensions

Edited by Emeric Deutsch, Mar 23 2005