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.

A308680 Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 4, 14, 19, 14, 5, 1, 0, 5, 22, 39, 36, 20, 6, 1, 0, 6, 34, 72, 85, 60, 27, 7, 1, 0, 8, 50, 128, 180, 160, 92, 35, 8, 1, 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1, 0, 12, 104, 354, 680, 845, 720, 434, 184, 54, 10, 1
Offset: 0

Views

Author

Alois P. Heinz, Aug 29 2019

Keywords

Comments

For fixed k > 0, T(n,k) ~ exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019
T is the convolution triangle of A000009 (see A357368). - Peter Luschny, Oct 19 2022

Examples

			T(4,1) = 2: 3a1a, 4a.
T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.
T(4,4) = 1: 1a1b1c1d.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  3,  8,   9,   4,   1;
  0,  4, 14,  19,  14,   5,   1;
  0,  5, 22,  39,  36,  20,   6,   1;
  0,  6, 34,  72,  85,  60,  27,   7,  1;
  0,  8, 50, 128, 180, 160,  92,  35,  8, 1;
  0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000009 (for n>0), A327380, A327381, A327382, A327383, A327384, A327385, A327386, A327387, A327388.
Main diagonal and lower diagonals give: A000012, A001477, A000096.
Row sums give A304969.
T(2n,n) gives A324595.
Cf. A060642, A286335, A325915.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
T:= proc(n, k) option remember;
      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, b(n)),
          (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jan 31 2021
# Uses function PMatrix from A357368.
PMatrix(10, A000009); # Peter Luschny, Oct 19 2022
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,    b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A286335(n,k-i).
Sum_{k=1..n} k * T(n,k) = A325915(n).
G.f. of column k: (-1 + Product_{j>=1} (1 + x^j))^k. - Alois P. Heinz, Jan 29 2021

A341223 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^5.

Original entry on oeis.org

1, 10, 55, 225, 765, 2287, 6215, 15680, 37265, 84300, 182933, 383070, 777705, 1536490, 2963120, 5592060, 10349465, 18817760, 33665870, 59341785, 103176877, 177131330, 300530125, 504318530, 837632700, 1377874861, 2246061540, 3630059510, 5819556060, 9258393655, 14622472250
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

Links

Crossrefs

Column k=5 of A060642.
Cf. A000041, A023004, A048574, A327383, A341221, A341222, A341225, A341226, A341227, A341228.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..35);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax = 35; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

A341387 Expansion of (-1 + Product_{k>=1} (1 + x^k)^k)^5.

Original entry on oeis.org

1, 10, 65, 320, 1330, 4872, 16255, 50335, 146775, 407045, 1082000, 2773045, 6884650, 16620225, 39135280, 90113553, 203347645, 450516450, 981491380, 2105504205, 4452798556, 9293254605, 19158353285, 39044262235, 78719105560, 157112112293, 310599279105
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Links

Crossrefs

Cf. A026007, A321950, A327383, A341384, A341385, A341386, A341388, A341390, A341391.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..31);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 31; CoefficientList[Series[(-1 + Product[(1 + x^k)^k, {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

A341244 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^5.

Original entry on oeis.org

1, 0, 5, 5, 15, 25, 45, 80, 125, 210, 321, 500, 745, 1110, 1620, 2326, 3315, 4660, 6500, 8955, 12261, 16640, 22425, 29990, 39870, 52701, 69230, 90460, 117620, 152225, 196066, 251455, 321195, 408710, 518060, 654317, 823690, 1033535, 1292690, 1611970, 2004462, 2485605
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

Crossrefs

Cf. A000700, A001483, A022600, A327383, A338463, A341223, A341241, A341243, A341245, A341246, A341247, A341251.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..46);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax = 46; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

Formula

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^5.

A341366 Expansion of (1 / theta_4(x) - 1)^5 / 32.

Original entry on oeis.org

1, 10, 60, 275, 1060, 3612, 11210, 32310, 87665, 226130, 558684, 1329720, 3062905, 6853310, 14941330, 31820642, 66343150, 135659570, 272496680, 538427720, 1047788137, 2010303890, 3806292130, 7118038360, 13157217715, 24055170690, 43527162380, 77994164515, 138463246700
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Crossrefs

Cf. A002448, A004406, A014968, A015128, A327383, A338223, A340481, A341223, A341364, A341365, A341367, A341368, A341369, A341370.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..33);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^5/32, {x, 0, nmax}], x] // Drop[#, 5] &
nmax = 33; CoefficientList[Series[(1/32) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

Formula

G.f.: (1/32) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^5.
Showing 1-5 of 5 results.

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