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 1-7 of 7 results.

A183240 Sums of the squares of multinomial coefficients.

Original entry on oeis.org

1, 1, 5, 46, 773, 19426, 708062, 34740805, 2230260741, 180713279386, 18085215373130, 2188499311357525, 315204533416762046, 53270712928769375885, 10441561861586014363349, 2349364090881443819316871, 601444438364480313663234821, 173817677082622796179263021770
Offset: 0

Views

Author

Paul D. Hanna, Jan 03 2011

Keywords

Comments

Equals sums of the squares of terms in rows of the triangle of multinomial coefficients (A036038).
Ignoring initial term, equals the logarithmic derivative of A183241; A183241 is conjectured to consist entirely of integers.
More generally, let {M(n,k), n>=0} be the sums of the k-th powers of the multinomial coefficients where k>=0 (see table A183610), then:
Sum_{n>=0} M(n,k)*x^n/n!^k = Product_{n>=1} 1/(1-x^n/n!^k).

Examples

			G.f.: A(x) = 1 + x + 5*x^2/2!^2 + 46*x^3/3!^2 + 773*x^4/4!^2 +...
A(x) = 1/((1-x)*(1-x^2/2!^2)*(1-x^3/3!^2)*(1-x^4/4!^2)*...).
...
After the initial term a(0)=1, the next several terms are
a(1) = 1^2 = 1,
a(2) = 1^2 + 2^2 = 5,
a(3) = 1^2 + 3^2 + 6^2 = 46,
a(4) = 1^2 + 4^2 + 6^2 + 12^2 + 24^2 = 773,
a(5) = 1^2 + 5^2 + 10^2 + 20^2 + 30^2 + 60^2 + 120^2 = 19426,
and continue with the sums of squares of the terms in triangle A036038.

Links

Crossrefs

Cf. A036038, A005651, A183235, A183236, A183237, A183238; A183241.
Cf. A183610 (table of sums of powers of multinomial coefficients).

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i))/i!^2+b(n, i-1))
    end:
a:= n-> n!^2*b(n$2):
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 11 2019
  • Mathematica
    t=Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i], Length[#1] > Length[#2] &]], {1}], {i, 30}]^2; Plus@@@t (* From Tony D. Noe *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1,
     b[n - i, Min[n - i, i]]/i!^2 + b[n, i - 1]];
a[n_] := n!^2 b[n, n];
a /@ Range[0, 21] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
  • PARI
    {a(n)=n!^2*polcoeff(1/prod(k=1,n,1-x^k/k!^2 +x*O(x^n)),n)}

Formula

G.f.: Sum_{n>=0} a(n)*x^n/n!^2 = Product_{n>=1} 1/(1-x^n/n!^2).
a(n) ~ c * (n!)^2, where c = Product_{k>=2} 1/(1-1/(k!)^2) = 1.37391178018464563291052028168404977854977270679629932106310942272080844... . - Vaclav Kotesovec, Feb 19 2015

Extensions

Terms following a(7) computed by T. D. Noe.

A183235 Sums of the cubes of multinomial coefficients.

Original entry on oeis.org

1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, 75263273895385, 54867365927680618, 54868847079435960134, 73030508546599681432983, 126197144644287414997433576, 277255161467330877411064074059
Offset: 0

Views

Author

Paul D. Hanna, Jan 04 2011

Keywords

Comments

Equals sums of the cubes of terms in rows of the triangle of multinomial coefficients (A036038).
Ignoring initial term, equals the logarithmic derivative of A182963.

Examples

			G.f.: A(x) = 1 + x + 9*x^2/2!^3 + 244*x^3/3!^3 + 15833*x^4/4!^3 +...
A(x) = 1/((1-x)*(1-x^2/2!^3)*(1-x^3/3!^3)*(1-x^4/4!^3)*...).
...
After the initial term a(0)=1, the next few terms are
a(1) = 1^3 = 1,
a(2) = 1^3 + 2^3 = 9,
a(3) = 1^3 + 3^3 + 6^3 = 244,
a(4) = 1^3 + 4^3 + 6^3 + 12^3 + 24^3 = 15833,
a(5) = 1^3 + 5^3 + 10^3 + 20^3 + 30^3 + 60^3 + 120^3 = 1980126, ...;
and continue with the sums of cubes of the terms in triangle A036038.

Links

Crossrefs

Cf. A036038, A005651, A183240, A183236, A183237, A183238; A182963.

Programs

  • PARI
    {a(n)=n!^3*polcoeff(1/prod(k=1, n, 1-x^k/k!^3 +x*O(x^n)), n)}

Formula

G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = Product_{n>=1} 1/(1 - x^n/n!^3).
a(n) ~ c * (n!)^3, where c = Product_{k>=2} 1/(1-1/(k!)^3) = 1.14825648754771664323845829539510031170864046029463094659207423270573478812675... . - Vaclav Kotesovec, Feb 19 2015

Extensions

Examples added and name changed by Paul D. Hanna, Jan 05 2011

A183236 Sums of multinomial coefficients to the 4th power.

Original entry on oeis.org

1, 1, 17, 1378, 354065, 221300626, 286871431922, 688780254549829, 2821284379712638737, 18510450092641988146882, 185104666826030540618018642, 2710117456989714966261367339909, 56196998736058707145628074314226034
Offset: 0

Views

Author

Paul D. Hanna, Jan 04 2011

Keywords

Comments

Equals sums of the 4th power of terms in rows of the triangle of multinomial coefficients (A036038).

Examples

			G.f.: A(x) = 1 + x + 17*x^2/2!^4 + 1378*x^3/3!^4 + 354065*x^4/4!^4 +...
A(x) = 1/((1-x)*(1-x^2/2!^4)*(1-x^3/3!^4)*(1-x^4/4!^4)*...).

Links

Crossrefs

Cf. A036038, A005651, A183240, A183235, A183237, A183238.

Programs

  • PARI
    {a(n)=n!^4*polcoeff(1/prod(k=1, n, 1-x^k/k!^4 +x*O(x^n)), n)}

Formula

G.f.: Sum_{n>=0} a(n)*x^n/n!^4 = Product_{n>=1} 1/(1 - x^n/n!^4).
a(n) ~ c * (n!)^4, where c = Product_{k>=2} 1/(1-1/(k!)^4) = 1.067493570155257423039762074691753715853526744464586468822554194836450214299287... . - Vaclav Kotesovec, Feb 19 2015

A183237 Sums of multinomial coefficients to the 5th power.

Original entry on oeis.org

1, 1, 33, 8020, 8220257, 25688403126, 199758931567152, 3357348771315829641, 110013706232123658318433, 6496199364012472451887572970, 649619955166586474874295658148158, 104621943411970982740307507415589286391
Offset: 0

Views

Author

Paul D. Hanna, Jan 04 2011

Keywords

Comments

Equals sums of the 5th power of terms in rows of the triangle of multinomial coefficients (A036038).

Examples

			G.f.: A(x) = 1 + x + 33*x^2/2!^5 + 8020*x^3/3!^5 + 8220257*x^4/4!^5 +...
A(x) = 1/((1-x)*(1-x^2/2!^5)*(1-x^3/3!^5)*(1-x^4/4!^5)*...).

Links

Crossrefs

Cf. A036038, A005651, A183240, A183235, A183236, A183238.

Programs

  • PARI
    {a(n)=n!^5*polcoeff(1/prod(k=1, n, 1-x^k/k!^5 +x*O(x^n)), n)}

Formula

G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = Product_{n>=1} 1/(1 - x^n/n!^5).
a(n) ~ c * (n!)^5, where c = Product_{k>=2} 1/(1-1/(k!)^5) = 1.03239096052278897179685563337623849923796538921602982416328969955606263213989... . - 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).

A215910 a(n) = sum of the n-th power of the multinomial coefficients in row n of triangle A036038.

Original entry on oeis.org

1, 1, 5, 244, 354065, 25688403126, 141528428949437282, 83257152559805973052807833, 7012360438832401192319979008881500417, 109324223115831487504443410090345278639832867784010, 396327911646787133737309113762487915762995734538047874429637296650
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2012

Keywords

Examples

			The sums of the n-th power of multinomial coefficients in row n of triangle A036038 begin:
a(1) = 1^1 = 1;
a(2) = 1^2 + 2^2 = 5;
a(3) = 1^3 + 3^3 + 6^3 = 244;
a(4) = 1^4 + 4^4 + 6^4 + 12^4 + 24^4 = 354065;
a(5) = 1^5 + 5^5 + 10^5 + 20^5 + 30^5 + 60^5 + 120^5 = 25688403126;
a(6) = 1^6 + 6^6 + 15^6 + 20^6 + 30^6 + 60^6 + 90^6 + 120^6 + 180^6 + 360^6 + 720^6 = 141528428949437282;
a(7) = 1^7 + 7^7 + 21^7 + 35^7 + 42^7 + 105^7 + 140^7 + 210^7 + 210^7 + 420^7 + 630^7 + 840^7 + 1260^7 + 2520^7 + 5040^7 = 83257152559805973052807833; ...
which also form a logarithmic generating function of an integer series:
L(x) = x + 5*x^2/2 + 244*x^3/3 + 354065*x^4/4 + 25688403126*x^5/5 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 84*x^3 + 88602*x^4 + 5137769389*x^5 +...+ A215911(n)*x^n +...

Links

Crossrefs

Cf. A183610, A036038, A036740, A005651, A183240, A183235, A183236, A183237, A183238; A215911.
Cf. A326321.

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-> n!^n*b(n$3):
seq(a(n), n=0..12);  # 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_] := n!^n b[n, n, n];
a /@ Range[0, 12] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)
  • PARI
    {a(n)=n!^n*polcoeff(1/prod(m=1, n, 1-x^m/m!^n +x*O(x^n)), n)}
for(n=1,15,print1(a(n),", "))

Formula

a(n) = [x^n/n!^n] * Product_{k=1..n} 1/(1 - x^k/k!^n) for n>=1, with a(0)=1.
Logarithmic derivative of A215911, ignoring the initial term a(0).
a(n) ~ (n!)^n = A036740(n). - Vaclav Kotesovec, Feb 19 2015
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Feb 19 2015

A183239 G.f.: exp( Sum_{n>=1} A005651(n)*x^n/n ), where A005651 gives the sums of multinomial coefficients.

Original entry on oeis.org

1, 1, 2, 5, 17, 69, 352, 2077, 14505, 114354, 1023839, 10130051, 110878314, 1320375213, 17086334702, 237832320231, 3552995476517, 56590659564489, 958653346775294, 17192978984630744, 325681548343314833, 6494280460641306608
Offset: 0

Views

Author

Paul D. Hanna, Jan 03 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 69*x^5 + 352*x^6 +...
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 47*x^4/4 + 246*x^5/5 + 1602*x^6/6 + 11481*x^7/7 + 95503*x^8/8 +...+ A005651(n)*x^n/n +...

Links

Crossrefs

Cf. A005651, A183238, A183241.

Programs

  • PARI
    {a(n)=polcoeff(exp(intformal(1/x*(-1+serlaplace(1/prod(k=1,n+1,1-x^k/k!+O(x^(n+2))))))),n)}

Formula

a(n) ~ c * (n-1)!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264... . - Vaclav Kotesovec, Feb 19 2015
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