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 1-9 of 9 results.

A333155 Decimal expansion of a constant related to the asymptotics of A268188 and A333153.

Original entry on oeis.org

5, 9, 3, 2, 4, 2, 2, 1, 5, 0, 0, 3, 3, 6, 9, 1, 2, 7, 1, 8, 4, 1, 3, 7, 6, 1, 7, 3, 3, 0, 2, 5, 5, 9, 5, 4, 1, 1, 0, 9, 9, 5, 9, 5, 4, 9, 6, 2, 7, 9, 5, 7, 4, 2, 9, 0, 6, 0, 2, 4, 5, 7, 8, 6, 0, 4, 5, 3, 5, 9, 2, 2, 3, 8, 5, 4, 6, 8, 1, 3, 3, 3, 3, 2, 5, 5, 0, 4, 8, 0, 7, 2, 0, 2, 8, 1, 9, 6, 6, 3, 9, 7, 1, 0, 7, 1
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

Examples

			0.5932422150033691271841376173302559541109959549627957429060245786...

Crossrefs

Cf. A003114, A268188, A333141, A333151, A333152.
Cf. A003106, A333153, A333154.

Programs

  • Maple
    evalf(sqrt(15) * log((sqrt(5) + 1)/2) / Pi, 120);
  • Mathematica
    RealDigits[Sqrt[15]*Log[GoldenRatio]/Pi, 10, 105][[1]]

Formula

Equals sqrt(15) * log(phi) / Pi, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
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) ~ A333155^m * phi^(1/2) * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2)).
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) ~ A333155^m * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)).

A268187 Triangle read by rows: T(n,k) is the number of no-leg partitions of n having Durfee square of size k (n >= 1, 1 <= k <= floor(sqrt(n))). Also, number of no-arm partitions of n having Durfee square of size k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 4, 1, 1, 4, 2, 1, 5, 3, 1, 5, 4, 1, 6, 5, 1, 6, 7, 1, 7, 8, 1, 1, 7, 10, 1, 1, 8, 12, 2, 1, 8, 14, 3, 1, 9, 16, 5, 1, 9, 19, 6, 1, 10, 21, 9, 1, 10, 24, 11, 1, 11, 27, 15, 1, 11, 30, 18, 1, 1, 12, 33, 23, 1
Offset: 1

Views

Author

Emeric Deutsch, Jan 29 2016

Keywords

Comments

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.

Examples

			T(9,2) = 3 because we have [7,2], [6,3], and [5,4].
Triangle begins:
  1;
  1;
  1;
  1, 1;
  1, 1;
  1, 2;
  1, 2;
  1, 3;
  1, 3, 1;
  1, 4, 1;
  1, 4, 2;
  1, 5, 3;
  1, 5, 4;
  1, 6, 5;
  1, 6, 7;
  1, 7, 8, 1;
  ...

Links

Crossrefs

Cf. A003114, A115994, A268188, A327710, A328346.

Programs

  • Maple
    G := add(t^k*x^(k^2)/mul(1-x^i, i = 1 .. k), k = 0 .. 80): Gser := simplify(series(G, x = 0,40)): for n to 35 do P[n] := sort(coeff(Gser, x, n)) end do: for n to 35 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
# 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:
T:= (n, k)-> b(n-k^2, k):
seq(seq(T(n, k), k=1..floor(sqrt(n))), n=1..30); # Alois P. Heinz, Jan 30 2016
  • Mathematica
    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]]]]; T[n_, k_] := b[n-k^2, k]; Table[T[n, k], {n, 1, 30}, {k, 1, Floor[Sqrt[n]]}] // Flatten (* Jean-François Alcover, Dec 10 2016 after Alois P. Heinz *)
  • PARI
    T(n, k) = polcoef(1/prod(j=1, k, 1-x^j+x*O(x^n)), n-k*k);
tabf(nn) = for(n=1, nn, for(k=1, sqrtint(n), print1(T(n, k), ", ")); print) \\ Seiichi Manyama, Oct 14 2019

Formula

G.f.: G(t,x) = Sum_{k>=0} ( t^k*x^(k^2)/Product_{i=1..k} (1-x^i) ).
Sum_{k>=0} T(n,k) = A003114(n).
Sum_{k>=1} k * T(n,k) = A268188(n).
Sum_{k>=0} k! * T(n,k) = A327710(n). - Alois P. Heinz, Feb 25 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

Views

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

Links

Crossrefs

Cf. A003114, A268188.

Programs

  • Maple
    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
  • Mathematica
    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]

Formula

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

Views

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

Links

Crossrefs

Cf. A003114, A268188, A333141.

Programs

  • Maple
    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
  • Mathematica
    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]

Formula

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

Views

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

Links

Crossrefs

Cf. A003106, A268188, A333154.

Programs

  • Maple
    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
  • Mathematica
    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]

Formula

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.

A333180 G.f.: Sum_{k>=1} (k * x^(k^2) * Product_{j=1..k} (1 + x^j)).

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 2, 2, 0, 3, 3, 3, 6, 3, 3, 3, 4, 4, 4, 8, 8, 8, 8, 8, 4, 9, 9, 5, 10, 10, 15, 15, 15, 15, 15, 15, 16, 16, 11, 17, 17, 18, 24, 24, 24, 30, 30, 30, 30, 31, 31, 31, 32, 26, 33, 34, 41, 41, 42, 49, 49, 56, 56, 56, 64, 64, 57, 65, 58, 59, 67, 68
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 10 2020

Keywords

Links

Crossrefs

Cf. A268188, A306734, A333181.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[n*x^(n^2)*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 0; Do[p = Expand[p*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += k*p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ c * A333198^sqrt(n), where c = 0.3836313809149103736315...
Limit_{n->infinity} a(n) / A333181(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...

A327710 Number of compositions of n into distinct parts such that the difference between any two parts is at least two.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 5, 5, 7, 13, 15, 21, 29, 35, 43, 55, 87, 99, 137, 173, 235, 277, 363, 429, 545, 755, 895, 1135, 1443, 1827, 2285, 2837, 3463, 4285, 5199, 6309, 8237, 9755, 12091, 14743, 18351, 22251, 27833, 33125, 40819, 49045, 59691, 70869, 86033, 106163
Offset: 0

Views

Author

Alois P. Heinz, Feb 24 2020

Keywords

Comments

All terms are odd.

Examples

			a(9) = 13: 135, 153, 315, 351, 513, 531, 36, 63, 27, 72, 18, 81, 9.

Links

Crossrefs

Cf. A003114, A032020, A268187, A268188, A328222.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
a:= n-> add(k!*b(n-k^2, k), k=0..floor(sqrt(n))):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := Sum[k!*b[n - k^2, k], {k, 0, Floor[Sqrt[n]]}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>=0} k! * A268187(n,k).
G.f.: Sum_{k>=0} k! * x^(k^2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Dec 04 2020

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

Views

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

Comments

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.

Links

Crossrefs

Cf. A003114, A268188, A333141, A333151.

Programs

  • Maple
    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
  • Mathematica
    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]

Formula

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

Views

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

Comments

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.

Links

Crossrefs

Cf. A003106, A268188, A333153.

Programs

  • Maple
    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
  • Mathematica
    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]

Formula

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.
Showing 1-9 of 9 results.

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

Content is available under The OEIS End-User License Agreement.