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.

Showing 1-5 of 5 results.

A237832 Number of partitions of n such that (greatest part) - (least part) = number of parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 4, 10, 8, 13, 15, 22, 22, 34, 36, 51, 58, 75, 85, 116, 130, 165, 194, 244, 281, 355, 409, 505, 591, 718, 839, 1022, 1186, 1425, 1668, 1994, 2319, 2765, 3213, 3805, 4429, 5214, 6052, 7119, 8243, 9645, 11169, 13026, 15046, 17511
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2014

Keywords

Examples

			a(6) = 2 counts these partitions:  4+2, 4+1+1.

Links

Crossrefs

Cf. A237830, A237831, A237833, A237834.

Programs

  • Mathematica
    z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*1/(1-x)*sum(k=1, N, (-1)^(k-1)*(k*(1-x)*x^(k*(3*k-1)/2)*(1+x^k)-x^(3*k*(k-1)/2+1)*(1-x^(2*k)))))) \\ Seiichi Manyama, May 20 2023

Formula

A237830(n) + a(n) + A237833(n) = A000041(n). - R. J. Mathar, Nov 24 2017
G.f.: (1/Product_{k>=1} (1-x^k)) * (1/(1-x)) * Sum_{k>=1} (-1)^(k-1) * ( k * (1-x) * x^(k*(3*k-1)/2) * (1+x^k) - x^(3*k*(k-1)/2+1) * (1-x^(2*k)) ) - Seiichi Manyama, May 20 2023

A237831 Number of partitions of n such that (greatest part) - (least part) <= number of parts.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 12, 18, 23, 32, 40, 57, 70, 94, 120, 157, 196, 256, 318, 408, 508, 640, 792, 996, 1223, 1518, 1863, 2296, 2798, 3432, 4162, 5070, 6130, 7422, 8936, 10777, 12916, 15500, 18522, 22136, 26348, 31376, 37222, 44160, 52236, 61756, 72824, 85847
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2014

Keywords

Examples

			a(6) = 10 counts all the 11 partitions of 6 except 4+1+2.

Links

Crossrefs

Cf. A109083, A237830, A237832, A237833, A237834.

Programs

  • Mathematica
    z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*k*(x^(k*(3*k-1)/2)+x^(k*(3*k+1)/2)))) \\ Seiichi Manyama, May 20 2023

Formula

a(n) + A237833(n) = A000041(n). - R. J. Mathar, Nov 24 2017
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * k * ( x^(k*(3*k-1)/2) + x^(k*(3*k+1)/2) ). (See Andrews' preprint.) - Seiichi Manyama, May 20 2023

A237830 Number of partitions of n such that (greatest part) - (least part) < number of parts.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 47, 62, 81, 105, 135, 174, 222, 282, 357, 450, 565, 707, 880, 1093, 1353, 1669, 2052, 2517, 3077, 3753, 4565, 5539, 6704, 8097, 9755, 11730, 14075, 16854, 20142, 24029, 28611, 34009, 40355, 47807, 56542, 66772, 78728
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2014

Keywords

Examples

			a(6) = 8 counts these partitions:  6, 3+3, 4+1+1, 3+2+1, 2+2+2, 3+1+1+1, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1.

Links

Crossrefs

Cf. A000070, A237831-A237834.

Programs

  • Mathematica
    z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
  • PARI
    my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*x/(1-x)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k-1)/2)*(1-x^(2*k)))) \\ Seiichi Manyama, May 20 2023

Formula

a(n) + A237834(n) = A000041(n). - R. J. Mathar, Nov 24 2017
G.f.: (1/Product_{k>=1} (1-x^k)) * (x/(1-x)) * Sum_{k>=1} (-1)^(k-1) * x^(3*k*(k-1)/2) * (1-x^(2*k)). (See Andrews' preprint.) - Seiichi Manyama, May 20 2023

A237834 Number of partitions of n such that (greatest part) - (least part) >= number of parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 7, 10, 15, 20, 30, 39, 54, 71, 96, 123, 163, 208, 270, 342, 437, 548, 695, 865, 1083, 1341, 1666, 2048, 2527, 3089, 3784, 4604, 5606, 6786, 8222, 9907, 11940, 14331, 17196, 20554, 24563, 29252, 34820, 41327, 49016, 57982, 68545, 80833
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2014

Keywords

Examples

			a(7) = 4 counts these partitions:  6+1, 5+2, 5+1+1, 4+2+1.

Links

Crossrefs

Cf. A237830, A237831, A237832, A237833.

Programs

  • Mathematica
    z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
Table[Count[IntegerPartitions[n],?(#[[1]]-#[[-1]]>=Length[#]&)],{n,50}] (* _Harvey P. Dale, Jul 21 2023 *)

Formula

A237830(n)+a(n) = A000041(n). - R. J. Mathar, Nov 24 2017

A275633 Andrews's shadow difference function D_3(q).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 20, 31, 41, 56, 74, 101, 129, 172, 219, 284, 363, 464, 581, 738, 924, 1155, 1435, 1785, 2199, 2717, 3332, 4084, 4987, 6076, 7375, 8949, 10817, 13051, 15706, 18877, 22622, 27078, 32332, 38545, 45870, 54496, 64618, 76525, 90463, 106788, 125863, 148145, 174106
Offset: 0

Views

Author

N. J. A. Sloane, Aug 09 2016

Keywords

Comments

Agrees with A237833 just for n <= 21.

Links

Crossrefs

Cf. A098151, A237833, A275632.

Programs

  • Maple
    F:=(a,q,n)->mul(1-a*q^i,i=0..n-1); # This is (a;q)_n
M:=15;
# A098151:
THETA3:=(add((-1)^n*q^(3*n^2),n=-M..M)) /(add((-1)^n*q^(n^2),n=-M..M));
s1:=series(THETA3,q,80); seriestolist(%);
# A275632:
THETABAR3:=1+2*add( (F(q,q,n-1)*q^(n^2)) / (F(q^n,q,n)*(1-q^n)), n=1..M);
s2:=series(THETABAR3,q,80); seriestolist(%);
# A275633:
series((s1-s2)/8,q,80); seriestolist(%);

Formula

Equals (A098151-A275632)/8.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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