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.

A215911 G.f.: exp( Sum_{n>=1} A215910(n)*x^n/n ), where A215910(n) equals the sum of the n-th power of multinomial coefficients in row n of triangle A036038.

Original entry on oeis.org

1, 1, 3, 84, 88602, 5137769389, 23588076629522583, 11893878960703225919597767, 876545054865944028047877165082786426, 12147135901759930712215268630715086378214795245696, 39632791164678725520866813137932593902239710762044280903318659253
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 84*x^3 + 88602*x^4 + 5137769389*x^5 +...
such that the logarithm of the g.f. begins:
log(A(x)) = x + 5*x^2/2 + 244*x^3/3 + 354065*x^4/4 + 25688403126*x^5/5 + 141528428949437282*x^6/6 +...+ A215910(n)*x^n/n +...
where the coefficients A215910(n) begin:
A215910(1) = 1^1 = 1;
A215910(2) = 1^2 + 2^2 = 5;
A215910(3) = 1^3 + 3^3 + 6^3 = 244;
A215910(4) = 1^4 + 4^4 + 6^4 + 12^4 + 24^4 = 354065;
A215910(5) = 1^5 + 5^5 + 10^5 + 20^5 + 30^5 + 60^5 + 120^5 = 25688403126; ...
and equal the sums of the n-th power of multinomial coefficients in row n of triangle A036038.

Links

Crossrefs

Cf. A215910, A183239, A183241, A183241, A182963.

Programs

  • PARI
    {a(n)=local(L=sum(m=1,n,m!^m*polcoeff(1/prod(k=1, n, 1-x^k/k!^m +x*O(x^m)), m)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}
for(n=0,15,print1(a(n),", "))

Formula

a(n) ~ (n!)^n / n. - Vaclav Kotesovec, Feb 19 2015
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2 - 1) / exp(n^2 - 1/12). - Vaclav Kotesovec, Feb 19 2015

A183610 Rectangular table where T(n,k) is the sum of the n-th powers of the k-th row of multinomial coefficients in triangle A036038 for n>=0, k>=0, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 5, 1, 1, 9, 46, 47, 7, 1, 1, 17, 244, 773, 246, 11, 1, 1, 33, 1378, 15833, 19426, 1602, 15, 1, 1, 65, 8020, 354065, 1980126, 708062, 11481, 22, 1, 1, 129, 47386, 8220257, 221300626, 428447592, 34740805, 95503, 30
Offset: 0

Views

Author

Paul D. Hanna, Aug 11 2012

Keywords

Examples

			The table begins:
n=0: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...];
n=1: [1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, ...];
n=2: [1, 1, 5, 46, 773, 19426, 708062, 34740805, 2230260741, ...];
n=3: [1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, ...];
n=4: [1, 1, 17, 1378, 354065, 221300626, 286871431922, ...];
n=5: [1, 1, 33, 8020, 8220257, 25688403126, 199758931567152, ...];
n=6: [1, 1, 65, 47386, 194139713, 3033434015626, 141528428949437282, ...];
n=7: [1, 1, 129, 282124, 4622599553, 361140600078126, ...];
n=8: [1, 1, 257, 1686178, 110507041025, 43166813000390626, ...];
n=9: [1, 1, 513, 10097380, 2646977660417, 5169878244001953126, ...];
n=10:[1, 1, 1025, 60525226, 63465359844353, 619778904740009765626, ...];
...
The sums of the n-th power of terms in row k of triangle A036038 begin:
T(n,1) = 1^n,
T(n,2) = 1^n + 2^n,
T(n,3) = 1^n + 3^n + 6^n,
T(n,4) = 1^n + 4^n + 6^n + 12^n + 24^n,
T(n,5) = 1^n + 5^n + 10^n + 20^n + 30^n + 60^n + 120^n,
T(n,6) = 1^n + 6^n + 15^n + 20^n + 30^n + 60^n + 90^n + 120^n + 180^n + 360^n + 720^n, ...
Note that row n=0 forms the partition numbers A000041.

Links

Crossrefs

Cf. A036038, A000041, A005651, A183240, A183235, A183236, A183237, A183238, A215910 (main diagonal).

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
    end:
A:= (n, k)-> k!^n*b(k$2, n):
seq(seq(A(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Sep 11 2019
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, 1, b[n-i, Min[n-i, i], k]/i!^k + b[n, i-1, k]];
A[n_, k_] := k!^n b[k, k, n];
Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)
  • PARI
    {T(n,k)=k!^n*polcoeff(1/prod(m=1, k, 1-x^m/m!^n +x*O(x^k)), k)}
for(n=0,10,for(k=0,8,print1(T(n,k),", "));print(""))

Formula

G.f. of row n: Sum_{k>=0} T(n,k)*x^k/k!^n = Product_{j>=1} 1/(1 - x^j/j!^n).

A326321 Sum of the n-th powers of multinomials M(n; mu), where mu ranges over all compositions of n.

Original entry on oeis.org

1, 1, 5, 271, 395793, 28076306251, 150414812114874563, 86530666539373619904011413, 7177587537701279221012034803727966465, 110824376322428312270365608303690048162629868273811, 399431453468560513224979712848478555015392084082614167438553312275
Offset: 0

Views

Author

Alois P. Heinz, Sep 11 2019

Keywords

Examples

			a(2) = M(2; 2)^2 + M(2; 1,1)^2 = 1 + 4 = 5.

Links

Crossrefs

Main diagonal of A326322.
Cf. A215910.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-i, k)/i!^k, i=1..n))
    end:
a:= n-> n!^n*b(n$2):
seq(a(n), n=0..12);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(binomial(n, j)^k*b(j, k), j=0..n-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..10);
  • Mathematica
    b[n_, k_] := b[n, k] = If[n==0, 1, Sum[Binomial[n, j]^k b[j, k], {j, 0, n-1}]];
a[n_] := b[n, n];
a /@ Range[0, 10] (* Jean-François Alcover, Dec 03 2020, after 2nd Maple program *)

Formula

From Vaclav Kotesovec, Sep 14 2019: (Start)
a(n) ~ (n!)^n.
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12). (End)
a(n) = (n!)^n * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k!)^n). - Ilya Gutkovskiy, Jul 11 2020

A336295 a(n) = (n!)^n * [x^n] Product_{k>=1} 1/(1 - x^k/k^n).

Original entry on oeis.org

1, 1, 5, 251, 359200, 25822962624, 141766192358448256, 83301485967496541735457536, 7013555995366382867427754604471779328, 109330254486209621988088555707809713786027354619904, 396335044092985772297627538614627390881554195217999599121962369024
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 16 2020

Keywords

Links

Crossrefs

Cf. A007841, A215910, A249588, A249593, A269791, A269793, A269794.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i), k)*((i-1)!*binomial(n, i))^k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 27 2023
  • Mathematica
    Table[(n!)^n SeriesCoefficient[Product[1/(1 - x^k/k^n), {k, 1, n}], {x, 0, n}], {n, 0, 10}]

A336294 a(n) = (n!)^n * [x^n] Product_{k>=1} (1 + x^k/(k!)^n).

Original entry on oeis.org

1, 1, 1, 28, 257, 103126, 46667437282, 140776183474585, 38414859209967468545, 8006615289848673023223926602, 100856872226698664486645150126408916015626, 7425498079138047573566961707334890995112470771975
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 16 2020

Keywords

Crossrefs

Cf. A007837, A183229, A183230, A215910.

Programs

  • Mathematica
    Table[(n!)^n SeriesCoefficient[Product[(1 + x^k/(k!)^n), {k, 1, n}], {x, 0, n}], {n, 0, 11}]
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