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

A123975 Number of Garden of Eden partitions of n in Bulgarian Solitaire.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 147, 190, 243, 311, 394, 499, 627, 786, 980, 1220, 1510, 1865, 2294, 2816, 3443, 4202, 5110, 6203, 7507, 9067, 10923, 13135, 15755, 18865, 22540, 26885, 32001, 38032, 45112, 53430, 63171
Offset: 1

Views

Author

Vladeta Jovovic, Nov 23 2006

Keywords

Comments

a(n) gives the number of times n occurs in A225794. - Antti Karttunen, Jul 27 2013

Links

Crossrefs

Cf. A000041, A045943, A064173, A101198, A237831.
Cf. A227753, A225794, A226062.

Programs

  • Maple
    p:=product(1/(1-q^i), i=1..200)*sum((-1)^(r-1)*q^((3*r^2+3*r)/2), r=1..200):s:=series(p, q, 200): for j from 0 to 199 do printf(`%d,`,coeff(s, q, j)) od: # James Sellers, Nov 30 2006
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k+1)/2)))) \\ Seiichi Manyama, May 21 2023

Formula

a(n) = A064173(n) - A101198(n).
a(n) = Sum_{j>=1} (-1)^(j+1)*p(n-b(j)) where b(j) = 3*j*(j+1)/2 (A045943) and p(n) is the number of partitions of n (see A000041). See Hopkins & Sellers. - Michel Marcus, Sep 26 2018
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 19*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

Extensions

More terms from James Sellers, Nov 30 2006

A237833 Number of partitions of n such that (greatest part) - (least part) > number of parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 20, 31, 41, 56, 74, 101, 129, 172, 219, 284, 362, 463, 579, 735, 918, 1147, 1422, 1767, 2172, 2680, 3279, 4013, 4888, 5947, 7200, 8721, 10515, 12663, 15202, 18235, 21798, 26039, 31015, 36898, 43802, 51930, 61426, 72590
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2014

Keywords

Examples

			a(8) = 4 counts these partitions:  7+1, 6+2, 6+1+1, 5+2+1.

Links

Crossrefs

Cf. A237830, A237831, A237832, A237834.
Different from, but has the same beginning as, A275633.

Programs

  • Maple
    isA237833 := proc(p)
    if abs(p[1]-p[-1]) > nops(p) then
        return 1;
    else
        return 0;
    end if;
end proc:
A237833 := proc(n)
    local a,p;
    a := 0 ;
    p := combinat[firstpart](n) ;
    while true do
        a := a+isA237833(p) ;
        if nops(p) = 1 then
            break;
        end if;
        p := nextpart(p) ;
    end do:
    return a;
end proc:
seq(A237833(n),n=1..20) ; # R. J. Mathar, Nov 17 2017
  • 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)); concat([0, 0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^k*(k-1)*(x^(k*(3*k-1)/2)+x^(k*(3*k+1)/2))))) \\ Seiichi Manyama, May 20 2023

Formula

A237831(n) + a(n) = A000041(n). - R. J. Mathar, Nov 24 2017
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^k * (k-1) * ( 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
Showing 1-5 of 5 results.

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