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.

Previous Showing 11-20 of 20 results.

A002758 Number of bipartite partitions of n white objects and 9 black ones.

Original entry on oeis.org

30, 97, 267, 608, 1279, 2472, 4571, 8043, 13715, 22652, 36535, 57568, 89079, 135384, 202747, 299344, 436597, 629364, 897970, 1268634, 1776562, 2466961, 3399463, 4650218, 6318429, 8529869, 11446563, 15272827, 20269135, 26762094
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of ways to factor p^n*q^9 where p and q are distinct primes.
a(n) is the number of multiset partitions of the multiset {r^n, s^9}. - Joerg Arndt, Jan 01 2024

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 2.
  • 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

Crossrefs

Column 9 of A054225.
Cf. A005380.

Programs

  • Mathematica
    p = 2; q = 3; b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[p^n*q^9, p^n*q^9]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(30 + 37*x + 43*x^2 + 37*x^3 + 20*x^4 - 3*x^5 - 35*x^6 - 65*x^7 - 97*x^8 - 119*x^9 - 109*x^10 - 69*x^11 - 26*x^12 + 37*x^13 + 89*x^14 + 131*x^15 + 142*x^16 + 141*x^17 + 97*x^18 + 44*x^19 - 18*x^20 - 72*x^21 - 100*x^22 - 108*x^23 - 96*x^24 - 69*x^25 - 25*x^26 + 12*x^27 + 42*x^28 + 52*x^29 + 54*x^30 + 35*x^31 + 14*x^32 + 2*x^33 - 4*x^34 - 20*x^35 - 19*x^36 - 14*x^37 - 8*x^38 + 7*x^39 + 8*x^40 + 8*x^41 - 2*x^42 - 4*x^43 + x^44)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8) * (1-x^9)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

Formula

a(n) = if n <= 9 then A054225(9,n) else A054225(n,9). - Reinhard Zumkeller, Nov 30 2011
a(n) ~ n^(7/2) * exp(Pi*sqrt(2*n/3)) / (560*sqrt(2)*Pi^9). - Vaclav Kotesovec, Feb 01 2016

Extensions

Edited by Christian G. Bower, Jan 08 2004

A002759 Number of bipartite partitions of n white objects and 10 black ones.

Original entry on oeis.org

42, 139, 392, 907, 1941, 3804, 7128, 12693, 21893, 36535, 59521, 94664, 147794, 226524, 342006, 508866, 747753, 1085635, 1559725, 2218272, 3126541, 4368724, 6056705, 8333955, 11388614, 15460291, 20859497, 27979454, 37324367, 49529018
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of ways to factor p^n*q^10 where p and q are distinct primes.
a(n) is the number of multiset partitions of the multiset {r^n, s^10}. - Joerg Arndt, Jan 01 2024

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 2.
  • 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

Crossrefs

Column 10 of A054225.
Cf. A005380.

Programs

  • Mathematica
    p = 2; q = 3; b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[p^n*q^10, p^n*q^10]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(42 + 55*x + 72*x^2 + 68*x^3 + 55*x^4 + 22*x^5 - 21*x^6 - 72*x^7 - 126*x^8 - 178*x^9 - 222*x^10 - 203*x^11 - 169*x^12 - 81*x^13 + 15*x^14 + 125*x^15 + 209*x^16 + 286*x^17 + 303*x^18 + 299*x^19 + 219*x^20 + 107*x^21 - 4*x^22 - 117*x^23 - 208*x^24 - 263*x^25 - 257*x^26 - 232*x^27 - 151*x^28 - 69*x^29 + 29*x^30 + 92*x^31 + 130*x^32 + 145*x^33 + 143*x^34 + 97*x^35 + 48*x^36 - 2*x^37 - 39*x^38 - 48*x^39 - 58*x^40 - 41*x^41 - 31*x^42 - 19*x^43 - 4*x^44 + 19*x^45 + 21*x^46 + 20*x^47 + 13*x^48 - 4*x^49 - 9*x^50 - 10*x^51 + 2*x^52 + 4*x^53 - x^54)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8) * (1-x^9) * (1-x^10)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

Formula

a(n) = if n <= 10 then A054225(10,n) else A054225(n,10). - Reinhard Zumkeller, Nov 30 2011
a(n) ~ sqrt(3) * n^4 * exp(Pi*sqrt(2*n/3)) / (5600*Pi^10). - Vaclav Kotesovec, Feb 01 2016

Extensions

Edited by Christian G. Bower, Jan 08 2004

A002763 Number of bipartite partitions.

Original entry on oeis.org

4, 11, 26, 52, 98, 171, 289, 467, 737, 1131, 1704, 2515, 3661, 5246, 7430, 10396, 14405, 19760, 26884, 36269, 48583, 64614, 85399, 112170, 146526, 190362, 246099, 316621, 405556, 517224, 657012, 831320, 1048055, 1316611, 1648486, 2057324, 2559719, 3175309
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 11.
  • 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

Crossrefs

Cf. A082775, A129306.

Programs

  • Maple
    with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b((45*2^n)$2):
seq(a(n), n=0..50);  # Alois P. Heinz, May 26 2013
  • Mathematica
    b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[45*2^n, 45*2^n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[(4 - x - 3*x^2 + x^3) / ((1 - x)^3 * (1 + x)) / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2017 *)

Formula

a(n) = a(n-1) + A000041(n) + A000070(n) + A000291(n), for n>0 - Alford Arnold, Dec 10 2007
From Vaclav Kotesovec, Jan 07 2017: (Start)
G.f.: (4 - x - 3*x^2 + x^3) / ((1-x)^3 * (1+x)) * Product_{k>=1} 1/(1-x^k).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 3*sqrt(n)/(2*sqrt(2)*Pi^3).
(End)

Extensions

Extended beyond a(25) by Alois P. Heinz, May 26 2013

A031129 Number of proper factorizations of p1^n*p2^6, where p1 and p2 are distinct primes.

Original entry on oeis.org

10, 29, 76, 161, 322, 588, 1042, 1751, 2875, 4570, 7127, 10859, 16305, 24050, 35039, 50354, 71608, 100696, 140348, 193783, 265504, 360888, 487213, 653242, 870612, 1153321, 1519657, 1991688, 2597761, 3372106, 4358197, 5608417
Offset: 0

Views

Author

Jeff Burch

Keywords

Links

Crossrefs

Cf. A002755, A028422.

Formula

a(n) = A002755(n) - 1 = A028422(2^n*3^6). - Ray Chandler, May 01 2017

Extensions

Offset changed to 0 and more terms added by Ray Chandler, May 01 2017

A002762 Number of bipartite partitions.

Original entry on oeis.org

4, 9, 21, 40, 74, 125, 209, 330, 515, 778, 1160, 1690, 2439, 3457, 4857, 6735, 9264, 12607, 17040, 22826, 30391, 40165, 52788, 68938, 89589, 115778, 148957, 190714, 243184, 308746, 390539, 492071, 617900, 773175, 964443, 1199168, 1486724, 1837806
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 11.
  • 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

Programs

  • Maple
    nmax := 40:
gf := (n, m, k) -> 1/(product(product(1-x^r*y^t, t=k..m), r=0..n) * product(1-x^s, s=1..n)):
seq(coeff(coeff(series(series(gf(nmax, 6, 2), x, nmax+1), y, 7), y, 6), x, n), n=0..nmax); # Sean A. Irvine, Aug 14 2014

Extensions

More terms from Sean A. Irvine, Aug 14 2014

A002764 Number of bipartite partitions.

Original entry on oeis.org

7, 18, 44, 88, 169, 296, 507, 824, 1314, 2029, 3083, 4578, 6714, 9676, 13795, 19408, 27053, 37302, 51029, 69180, 93139, 124447, 165259, 218021, 286068, 373207, 484512, 625845, 804840, 1030369, 1313823, 1668466, 2111101, 2661365, 3343811, 4187191
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 11.
  • 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

Programs

  • Maple
    nmax := 35:
gf := (n, m, k) -> 1/(product(product(1-x^r*y^t, t=k..m), r=0..n) * product(1-x^s, s=1..n)):
seq(coeff(coeff(series(series(gf(nmax, 8, 2), x, nmax+1), y, 9), y, 8), x, n), n=0..nmax); # Sean A. Irvine, Aug 14 2014

Extensions

More terms from Sean A. Irvine, Aug 14 2014

A002765 Number of bipartite partitions.

Original entry on oeis.org

8, 23, 57, 119, 231, 415, 719, 1189, 1915, 2997, 4595, 6898, 10198, 14833, 21303, 30211, 42393, 58869, 81028, 110551, 149683, 201160, 268539, 356167, 469630, 615712, 803029, 1042051, 1345896, 1730473, 2215561, 2825037, 3588364, 4541036
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 11.
  • 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

Programs

  • Maple
    nmax := 35:
gf := (n, m, k) -> 1/(product(product(1-x^r*y^t, t=k..m), r=0..n) * product(1-x^s, s=1..n)):
seq(coeff(coeff(series(series(gf(nmax, 9, 2), x, nmax+1), y, 10), y, 9), x, n), n=0..nmax); # Sean A. Irvine, Aug 14 2014

Extensions

More terms from Sean A. Irvine, Aug 14 2014

A002766 Number of bipartite partitions.

Original entry on oeis.org

4, 10, 23, 45, 83, 142, 237, 377, 588, 892, 1330, 1943, 2804, 3982, 5595, 7768, 10686, 14555, 19674, 26371, 35112, 46424, 61015, 79705, 103579, 133883, 172243, 220551, 281212, 357043, 451592, 568997, 714424, 893921, 1114930, 1386187
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 19.
  • 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

Programs

  • Maple
    nmax := 20:
gf := (n, m, k) -> 1/(product(product(1-x^r*y^t, t=k..m), r=0..n) * product(1-x^s, s=1..n)):
seq(coeff(coeff(series(series(gf(nmax, 9, 3), x, nmax+1), y, 10), y, 9), x, n), n=0..nmax); # Sean A. Irvine, Aug 14 2014

Extensions

More terms from Sean A. Irvine, Aug 14 2014

A002767 Number of bipartite partitions.

Original entry on oeis.org

5, 12, 28, 54, 100, 170, 284, 450, 702, 1062, 1583, 2308, 3329, 4720, 6628, 9190, 12634, 17189, 23219, 31092, 41371, 54651, 71782, 93695, 121684, 157169, 202080, 258579, 329509, 418096, 528518, 665521, 835170, 1044408, 1301949, 1617830
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 26.
  • 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

Programs

  • Maple
    nmax := 20:
gf := (n, m, k) -> 1/(product(product(1-x^r*y^t, t=k..m), r=0..n) * product(1-x^s, s=1..n)):
seq(coeff(coeff(series(series(gf(nmax, 12, 4), x, nmax+1), y, 13), y, 12), x, n), n=0..nmax); # Sean A. Irvine, Aug 15 2014

Extensions

More terms from Sean A. Irvine, Aug 15 2014

A002768 Number of bipartite partitions.

Original entry on oeis.org

5, 13, 30, 59, 109, 187, 312, 497, 775, 1176, 1753, 2561, 3694, 5245, 7366, 10223, 14056, 19137, 25853, 34637, 46092, 60910, 80009, 104462, 135674, 175274
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 32.
  • 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

Programs

  • Maple
    nmax := 20:
gf := (n, m, k) -> 1/(product(product(1-x^r*y^t, t=k..m), r=0..n) * product(1-x^s, s=1..n)):
seq(coeff(coeff(series(series(gf(nmax, 15, 5), x, nmax+1), y, 16), y, 15), x, n), n=0..nmax); # Sean A. Irvine, Aug 15 2014
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