A181144
G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.
Original entry on oeis.org
1, 1, 1, 1, 9, 1, 1, 36, 36, 1, 1, 100, 419, 100, 1, 1, 225, 2699, 2699, 225, 1, 1, 441, 12138, 35052, 12138, 441, 1, 1, 784, 42865, 286206, 286206, 42865, 784, 1, 1, 1296, 127191, 1696820, 3932898, 1696820, 127191, 1296, 1, 1, 2025, 330903, 7958563
Offset: 0
G.f.: A(x,y) = 1 + (1+y)*x + (1+9*y+y^2)*x^2 + (1+36*y+36*y^2+y^3)*x^3 + (1+100*y+419*y^2+100*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^4*y + y^2)*x^2/2
+ (1 + 3^4*y + 3^4*y^2 + y^3)*x^3/3
+ (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4
+ (1 + 5^4*y + 10^4*y^2 + 10^4*y^3 + 5^4*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 9, 1;
1, 36, 36, 1;
1, 100, 419, 100, 1;
1, 225, 2699, 2699, 225, 1;
1, 441, 12138, 35052, 12138, 441, 1;
1, 784, 42865, 286206, 286206, 42865, 784, 1;
1, 1296, 127191, 1696820, 3932898, 1696820, 127191, 1296, 1;
1, 2025, 330903, 7958563, 36955542, 36955542, 7958563, 330903, 2025, 1;
1, 3025, 776688, 31205941, 261852055, 525079969, 261852055, 31205941, 776688, 3025, 1; ...
Note that column 1 forms the sum of cubes (A000537), and forms the squares of the triangular numbers.
Inverse binomial transform of columns begins:
[1];
[1, 8, 19, 18, 6];
[1, 35, 348, 1549, 3713, 5154, 4161, 1818, 333];
[1, 99, 2500, 27254, 161793, 589819, 1409579, 2282850, 2529900, 1893972, 917349, 259854, 32726]; ...
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{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,m,binomial(m,j)^4*y^j)*x^m/m)+O(x^(n+1))),n,x),k,y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
A166896
G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k] * x^n/n ), an integer series in x.
Original entry on oeis.org
1, 1, 2, 6, 16, 45, 142, 459, 1508, 5122, 17787, 62649, 223971, 811339, 2970032, 10974150, 40893393, 153512844, 580082454, 2205046961, 8427087958, 32362949488, 124837337235, 483508287359, 1879669861074, 7332469937755
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 39*x^4/4 + 126*x^5/5 + 477*x^6/6 + 1765*x^7/7 +...+ A166897(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3*x^k)*x^m/m)+x*O(x^n)), n)}
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m\2, binomial(m-k, k)^3*m/(m-k))*x^m/m)+x*O(x^n)), n)}
A166899
a(n) = Sum_{k=0..[n/2]} C(n-k,k)^4*n/(n-k), n>=1.
Original entry on oeis.org
1, 3, 25, 111, 456, 2697, 15961, 86247, 495781, 3003738, 17946798, 107667969, 660458787, 4081397547, 25274724105, 157744019799, 991384251102, 6254115981009, 39613066988527, 252017709962526, 1608980424431755
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 111*x^4/4 + 456*x^5/5 + 2697*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 38*x^4 + 137*x^5 + 646*x^6 + 3241*x^7 +...+ A166898(n)*x^n +...
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Table[Sum[Binomial[n - k, k]^4 *n/(n - k), {k, 0, Floor[n/2]}], {n, 1, 50}] (* G. C. Greubel, May 27 2016 *)
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a(n)=sum(k=0,n\2,binomial(n-k,k)^4*n/(n-k))
A196560
G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4 * x^k]^2 * x^n/n ).
Original entry on oeis.org
1, 1, 3, 20, 205, 2624, 24793, 283522, 3639005, 50426826, 740744940, 10801827249, 163698355616, 2554965416964, 40878247859612, 667841855292388, 11051724909284834, 185702751266940874, 3162454792706586691, 54508849210857505845
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 205*x^4 + 2624*x^5 + 24793*x^6 +...
where
log(A(x)) = (1 + x)^2*x + (1+2^4*x+x^2)^2*x^2/2 + (1+3^4*x+3^4*x^2+x^3)^2*x^3/3 + (1+4^4*x+6^4*x^2+4^4*x^3+x^4)^2*x^4/4 +...
More explicitly,
log(A(x)) = x + 5*x^2/2 + 52*x^3/3 + 733*x^4/4 + 11926*x^5/5 + 129944*x^6/6 +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^4*x^k)^2*x^m/m)+x*O(x^n)), n)}
Showing 1-4 of 4 results.
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