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.

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A103926 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 7.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 184, 297, 473, 734, 1127, 1696, 2526, 3707, 5388, 7737, 11018, 15532, 21731, 30147, 41538, 56813, 77234, 104317, 140120, 187139, 248680, 328769, 432664, 566759, 739297, 960315, 1242583, 1601645, 2057095, 2632724
Offset: 0

Views

Author

Wolfdieter Lang, Mar 24 2005

Keywords

Comments

See A103923 for other combinatorial interpretations of a(n).
Also the sum of binomial (D(p), 7) over partitions p of n+28, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

References

  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 90.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Crossrefs

Eighth column (m=7) of Fine-Riordan triangle A008951 and of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.
Cf. A000712 (all parts of two kinds).

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 7}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@7], {n,0,37}] (* Robert Price, Jul 29 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 + 28, 7];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Formula

G.f.: (product(1/(1-x^k), k=1..7)^2)*product(1/(1-x^j), j=8..infty).
a(n) = sum(A103924(n-7*j), j=0..floor(n/7)), n>=0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(7/2) * n^(5/2) / (4*sqrt(3) * 7! * Pi^7). - Vaclav Kotesovec, Aug 28 2015

A103927 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 8.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 299, 478, 744, 1147, 1732, 2591, 3817, 5573, 8036, 11496, 16276, 22878, 31879, 44129, 60630, 82807, 112353, 151616, 203415, 271558, 360648, 476793, 627389, 822104, 1072668, 1394199, 1805060, 2328653, 2993372, 3835068, 4897199
Offset: 0

Views

Author

Wolfdieter Lang, Mar 24 2005

Keywords

Comments

See A103923 for other combinatorial interpretations of a(n).
Also the sum of binomial (D(p), 8) over partitions p of n+36, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

References

  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 90.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Crossrefs

Ninth column (m=8) of Fine-Riordan triangle A008951 and of triangle A103923, i.e., the p_2(n, m) array of the Gupta et al. reference.
Cf. A000712 (all parts of two kinds).

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(add(d+
     `if`(d<9, d, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 11 2018
  • Mathematica
    nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 8}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@8], {n,0,39}] (* Robert Price, Jul 29 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 + 36, 8];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Formula

G.f.: (Product_{k=1..8} 1/(1-x^k))^2*Product_{j>=9} 1/(1-x^j).
a(n) = Sum_{j=0..floor(n/8)} A103924(n-8*j), n >= 0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^4 * n^3 / (4*sqrt(3) * 8! * Pi^8). - Vaclav Kotesovec, Aug 28 2015

A103928 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 9.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 480, 749, 1157, 1752, 2627, 3882, 5683, 8221, 11796, 16756, 23627, 33036, 45881, 63257, 86689, 118036, 159837, 215211, 288314, 384275, 509829, 673270, 885361, 1159357, 1512235, 1964897, 2543864, 3281686
Offset: 0

Views

Author

Wolfdieter Lang, Mar 24 2005

Keywords

Comments

See A103923 for other combinatorial interpretations of a(n).
Also the sum of binomial (D(p), 9) over partitions p of n+45, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

References

  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 91.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Crossrefs

Tenth column (m=9) of Fine-Riordan triangle A008951 and of triangle A103923, i.e., the p_2(n, m) array of the Gupta et al. reference.
Cf. A000712 (all parts of two kinds).

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 9}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@9], {n,0,37}] (* Robert Price, Jul 29 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 + 45, 9];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Formula

G.f.: (Product_{k=1..9} 1/(1-x^k))^2 * Product_{j>=10} 1/(1-x^j).
a(n) = Sum_{j=0..floor(n/9)} A103924(n-9*j), n >= 0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(9/2) * n^(7/2) / (4*sqrt(3) * 9! * Pi^9). - Vaclav Kotesovec, Aug 28 2015
Previous Showing 11-13 of 13 results.

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