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
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.
Cf.
A183610 (table of sums of powers of multinomial coefficients).
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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
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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 *)
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{a(n)=n!^2*polcoeff(1/prod(k=1,n,1-x^k/k!^2 +x*O(x^n)),n)}
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
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 +...
A182963
G.f.: A(x) = exp( Sum_{n>=1} A183235(n)*x^n/n ) where A183235 is the sums of the cubes of multinomial coefficients.
Original entry on oeis.org
1, 1, 5, 86, 4052, 400401, 71827456, 21068995258, 9429303819612, 6105894632883407, 5493030296624140330, 6644655430011095138676, 10523095865317003368417750, 21337870239129956669159151372
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 86*x^3 + 4052*x^4 + 400401*x^5 +...
log(A(x)) = x + 9*x^2/2 + 244*x^3/3 + 15833*x^4/4 + 1980126*x^5/5 + 428447592*x^6/6 + 146966837193*x^7/7 +...+ A183235(n)*x^n/n +...
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{a(n)=polcoeff(exp(intformal(1/x*(-1+serlaplace(serlaplace(serlaplace(1/prod(k=1, n+1, 1-x^k/k!^3+O(x^(n+2))))))))), n)}
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
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.
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{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),", "))
Showing 1-4 of 4 results.
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