A102420 -id:A102420 - cp's OEIS frontend

A102422 Number of partitions of n with k <= 5 parts and each part p <= 5.

1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 18, 19, 20, 20, 19, 18, 16, 14, 11, 9, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Thomas Wieder, Jan 09 2005

There are only 26 nonzero terms.
Contribution from Toby Gottfried, Feb 19 2009: (Start)
a(n) is the number of partitions of n+5 into exactly 5 parts with each part p: 1 <= p <= 6
i.e. the number of different ways to get a total of n+5 with 5 (normal, 6-sided) dice in any order (End)

			a(7)=11 because we can write 7=1+2+2+2 or 5+2 or 1+2+4 or 3+4 or 1+3+3 or 1+1+1+1+3 or 1+1+2+3 or 2+2+3 or 1+1+1+2+2 1+1+1+4 or 1+1+5.
A total of 8 comes from 1+1+1+1+4, 1+1+1+2+3, 1+1+2+2+2 and a(3) = 3 [8 = 3+5] [From _Toby Gottfried_, Feb 19 2009]

See A102420 for k=5 and p<=5.
Cf. A000041, A102420, A063746.
Contribution from Toby Gottfried, Feb 19 2009: (Start)
A102420 has the numbers for 4 dice
A063260 gives the number of permuted rolls of each possible total for any number of dice. (End)

G.f.: = 1+z+2*z^2+3*z^3+5*z^4+7*z^5+9*z^6+11*z^7+14*z^8+16*z^9+18*z^10+19*z^11+20*z^12+20*z^13+19*z^14+18*z^15+16*z^16+14*z^17+11*z^18+9*z^19 +7*z^20+5*z^21+3*z^22+2*z^23+z^24+z^25.

A102424 Number of partitions of n with each part p <= 5 and each part's multiplicity m <= 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 25, 30, 36, 43, 50, 58, 66, 75, 84, 94, 104, 114, 124, 135, 145, 156, 165, 175, 184, 193, 201, 208, 214, 220, 224, 228, 230, 231, 231, 230, 228, 224, 220, 214, 208, 201, 193, 184, 175, 165, 156, 145, 135, 124, 114, 104, 94, 84, 75, 66, 58, 50, 43, 36, 30, 25, 20, 16, 12, 9, 7, 5, 3, 2, 1, 1

Offset: 0

Thomas Wieder, Jan 09 2005

There are only 76 nonzero terms.

			a(7)=12 because we can write 7=1+1+1+1+1+2, 1+1+1+2+2, 1+2+2+2, 1+1+1+1+3, 1+1+2+3, 2+2+3, 1+3+3, 1+1+1+4, 1+2+4, 3+4, 1+1+5, 2+5. Not allowed are: 1+1+1+1+1+1+1, 16, 7.

Cf. A102420 = number of partitions of integer n with exactly k = 5 parts and each part p <= 5.

Cf. A000041, A102420.

g:=product(sum(z^(p*m),m=0..5),p=1..5): series(g,z=0,80);

nonzeroterms() = {my(res = vector(76)); forvec(x = vector(5, i, [0, 5]), c = x*[1..5]~; res[c+1]++); res} \\ David A. Corneth, Aug 22 2020

a(n) = a(75 - n). - David A. Corneth, Aug 22 2020

G.f.: Product_{m=1..5} Sum_{k=0..5} x^(j*k). - Joerg Arndt, Aug 23 2020

Edited by N. J. A. Sloane, Sep 15 2006

Missing term 23 inserted by David A. Corneth, Aug 22 2020

cp's OEIS Frontend

A102422 Number of partitions of n with k <= 5 parts and each part p <= 5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A102424 Number of partitions of n with each part p <= 5 and each part's multiplicity m <= 5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Formula

Extensions