A333155 -id:A333155 - cp's OEIS frontend

A268188 Sum of the sizes of the Durfee squares of all no-leg partitions of n (or of all no-arm partitions of n).

1, 1, 1, 3, 3, 5, 5, 7, 10, 12, 15, 20, 23, 28, 34, 43, 49, 61, 71, 87, 100, 120, 137, 164, 190, 221, 254, 298, 340, 396, 451, 520, 592, 679, 769, 883, 996, 1133, 1278, 1453, 1632, 1850, 2072, 2339, 2620, 2947, 3288, 3695, 4119, 4608, 5129, 5728, 6360, 7091, 7862

Offset: 1

Emeric Deutsch, Jan 29 2016

nonn

Given a partition P, the partition formed by the cells situated below the Durfee square of P is called the leg of P. Similarly, the partition formed by the cells situated to the right of the Durfee square of P is called the arm of P.
Conjecture: also a(n) is the number of parts in the partitions of n whose parts differ by at least 2. For example, for n = 5, these partitions are 5, 41 with 3 parts in all. George Beck, Apr 22 2017
Note added Apr 22 2017. George E. Andrews informed me that this is part of the common interpretation of the Rogers-Ramanujan identities. - George Beck

			a(9)=10 because the no-leg partitions of 9 are [9], [7,2], [6,3], [5,4], and [3,3,3] with sizes of Durfee squares 1,2,2,2, and 3, respectively.

George E. Andrews, "Partitions and Durfee Dissection", Amer. J. Math. 101(1979), 735-742.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A003114, A268187, A333141, A333151, A333152, A333155.

g := add(k*x^(k^2)/mul(1-x^i, i = 1 .. k), k = 1 .. 100): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 1 .. 55);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k*b(n-k^2, k), k=1..floor(sqrt(n))):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 30 2016

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]; a[n_] := Sum[k*b[n - k^2, k], {k, 1, Floor[Sqrt[n]]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

a(n) = Sum_{k>=1} k*A268187(n,k).
G.f.: g = Sum_{k>=1} (k*x^(k^2)/Product_{i=1..k}(1-x^i)).
a(n) ~ 3^(1/4) * log(phi) * phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2*Pi*n^(1/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 09 2020

A333141 G.f.: Sum_{k>=1} (k^2 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

Original entry on oeis.org

0, 1, 1, 1, 5, 5, 9, 9, 13, 22, 26, 35, 48, 57, 70, 88, 117, 135, 173, 207, 261, 304, 374, 433, 528, 628, 739, 864, 1032, 1198, 1416, 1639, 1914, 2212, 2569, 2949, 3433, 3920, 4511, 5150, 5925, 6732, 7720, 8736, 9969, 11284, 12823, 14444, 16395, 18457, 20836

Offset: 0

Vaclav Kotesovec, Mar 09 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003114, A268188.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k^2 * b(n-k^2, k), k=1..floor(sqrt(n))):
seq(a(n), n=0..50); # after Alois P. Heinz

nmax = 50; CoefficientList[Series[Sum[n^2 * x^(n^2) / Product[1 - x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(1/4), where c = A333155^2 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.076061100391958657489521534823556... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A333151 G.f.: Sum_{k>=1} (k^3 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

Original entry on oeis.org

0, 1, 1, 1, 9, 9, 17, 17, 25, 52, 60, 87, 122, 149, 184, 238, 337, 391, 517, 635, 825, 970, 1224, 1433, 1778, 2176, 2585, 3074, 3736, 4414, 5292, 6223, 7354, 8626, 10135, 11785, 13915, 16068, 18701, 21600, 25141, 28884, 33512, 38288, 44165, 50494, 57961

Offset: 0

Vaclav Kotesovec, Mar 09 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003114, A268188, A333141.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k^3 * b(n-k^2, k), k=1..floor(sqrt(n))):
seq(a(n), n=0..50);  # after Alois P. Heinz

nmax = 50; CoefficientList[Series[Sum[n^3*x^(n^2)/Product[1-x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(3/4), where c = A333155^3 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.04512265567211918167849606290245... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A333153 G.f.: Sum_{k>=1} (k * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 12, 12, 17, 20, 25, 28, 36, 39, 51, 57, 69, 79, 98, 108, 131, 148, 175, 196, 235, 260, 307, 344, 400, 450, 522, 581, 671, 751, 859, 957, 1097, 1218, 1385, 1543, 1744, 1940, 2193, 2428, 2735, 3033, 3400, 3763, 4215, 4654

Offset: 0

Vaclav Kotesovec, Mar 09 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003106, A268188, A333154.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k * b(n-k*(k+1), k), k=1..floor(sqrt(n))):
seq(a(n), n=0..60);  # after Alois P. Heinz

nmax = 60; CoefficientList[Series[Sum[n * x^(n*(n+1)) / Product[1 - x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * exp(2*Pi*sqrt(n/15)) / n^(1/4), where c = A333155 / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)) = 0.07923971705837122678006319599762... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A333152 G.f.: Sum_{k>=1} (k^4 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

Original entry on oeis.org

0, 1, 1, 1, 17, 17, 33, 33, 49, 130, 146, 227, 324, 405, 502, 664, 1017, 1179, 1613, 2031, 2721, 3220, 4166, 4921, 6204, 7840, 9379, 11352, 14028, 16882, 20520, 24511, 29286, 34864, 41401, 48741, 58417, 68144, 80207, 93698, 110325, 128124, 150436, 173424

Offset: 0

Vaclav Kotesovec, Mar 09 2020

nonn

In general, if m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k^2) / Product_{j=1..k} (1 - x^j)), then a(n) ~ r^m * phi^(1/2) * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2)), where r = A333155 = sqrt(15) * log(phi) / Pi = 0.59324221500336912718413761733... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003114, A268188, A333141, A333151.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k^4 * b(n-k^2, k), k=1..floor(sqrt(n))):
seq(a(n), n=0..50);  # after Alois P. Heinz

nmax = 50; CoefficientList[Series[Sum[n^4*x^(n^2)/Product[1-x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(5/4), where c = A333155^4 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.026768664197762321048783840410317... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A333154 G.f.: Sum_{k>=1} (k^2 * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 5, 5, 9, 9, 13, 13, 26, 26, 39, 48, 61, 70, 92, 101, 139, 157, 195, 229, 292, 326, 405, 464, 559, 634, 779, 870, 1047, 1188, 1406, 1604, 1888, 2127, 2493, 2823, 3271, 3683, 4283, 4802, 5525, 6221, 7112, 7992, 9137, 10210, 11625, 13013, 14734

Offset: 0

Vaclav Kotesovec, Mar 09 2020

nonn

In general, if m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)), then a(n) ~ r^m * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)), where r = A333155 = sqrt(15) * log(phi) / Pi = 0.59324221500336912718413761733... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003106, A268188, A333153.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k^2 * b(n-k*(k+1), k), k=1..floor(sqrt(n))):
seq(a(n), n=0..60);  # after Alois P. Heinz

nmax = 60; CoefficientList[Series[Sum[n^2 * x^(n*(n+1)) / Product[1 - x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(1/4), where c = A333155^2 / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)) = 0.04700834526394839955207674000683... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

cp's OEIS Frontend

A268188 Sum of the sizes of the Durfee squares of all no-leg partitions of n (or of all no-arm partitions of n).

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A333141 G.f.: Sum_{k>=1} (k^2 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

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A333151 G.f.: Sum_{k>=1} (k^3 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

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A333153 G.f.: Sum_{k>=1} (k * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)).

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A333152 G.f.: Sum_{k>=1} (k^4 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

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Programs

Maple

Mathematica

Formula

A333154 G.f.: Sum_{k>=1} (k^2 * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula