A166990
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
Original entry on oeis.org
1, 2, 7, 30, 147, 786, 4472, 26644, 164477, 1044258, 6782484, 44887236, 301782361, 2056250570, 14172792355, 98667874038, 692948001906, 4904403499992, 34951124337300, 250617829087656, 1807055528439771, 13095146839953030
Offset: 0
G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + 786*x^5 + 4472*x^6 +...
log(A(x)) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...
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a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/n, {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)
nmax = 30; Clear[a]; franel = RecurrenceTable[{n^2*a[n] == (7*n^2 - 7*n + 2)*a[n-1] + 8*(n-1)^2*a[n-2], a[1] == 2, a[2] == 10}, a, {n, 1, nmax}]; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[franel[[k]]*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 27 2024 *)
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{a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)*x^m/m)+x*O(x^n)),n)}
A218116
G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6 * 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, 33, 1, 1, 276, 276, 1, 1, 1300, 12695, 1300, 1, 1, 4425, 221495, 221495, 4425, 1, 1, 12201, 2185350, 11534720, 2185350, 12201, 1, 1, 29008, 14794261, 285715550, 285715550, 14794261, 29008, 1, 1, 61776, 76579851, 4276969276, 15781532964
Offset: 0
G.f.: A(x,y) = 1 + (1+y)*x + (1+33*y+y^2)*x^2 + (1+276*y+276*y^2+y^3)*x^3 + (1+1300*y+12695*y^2+1300*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^6*y + y^2)*x^2/2
+ (1 + 3^6*y + 3^6*y^2 + y^3)*x^3/3
+ (1 + 4^6*y + 6^6*y^2 + 4^6*y^3 + y^4)*x^4/4
+ (1 + 5^6*y + 10^6*y^2 + 10^6*y^3 + 5^6*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 33, 1;
1, 276, 276, 1;
1, 1300, 12695, 1300, 1;
1, 4425, 221495, 221495, 4425, 1;
1, 12201, 2185350, 11534720, 2185350, 12201, 1;
1, 29008, 14794261, 285715550, 285715550, 14794261, 29008, 1;
1, 61776, 76579851, 4276969276, 15781532964, 4276969276, 76579851, 61776, 1;
1, 120825, 324104715, 44480357175, 478591541712, 478591541712, 44480357175, 324104715, 120825, 1; ...
Note that column 1 forms the sum of fifth powers (A000539).
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{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^6*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(""))
A218117
G.f.: A(x) = exp( Sum_{n>=1} A005261(n)*x^n/n ) where A005261(n) = Sum_{k=0..n} C(n,k)^5.
Original entry on oeis.org
1, 2, 19, 198, 2961, 49566, 938322, 19083624, 412160478, 9305822076, 217855152321, 5251363667622, 129704365956114, 3269927116717728, 83893626609970281, 2185188966488265718, 57673989852987800966, 1539973309401567102832, 41544812360973818992909
Offset: 0
G.f.: A(x) = 1 + 2*x + 19*x^2 + 198*x^3 + 2961*x^4 + 49566*x^5 + 938322*x^6 +...
log(A(x)) = 2*x + 34*x^2/2 + 488*x^3/3 + 9826*x^4/4 + 206252*x^5/5 + 4734304*x^6/6 + 113245568*x^7/7 +...+ A005261(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^5)*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))
A218120
G.f.: A(x) = exp( Sum_{n>=1} A069865(n)/2*x^n/n ) where A069865(n) = Sum_{k=0..n} C(n,k)^6.
Original entry on oeis.org
1, 1, 17, 260, 7244, 214257, 7593707, 287419304, 11745920475, 503237634257, 22503750152879, 1039694201489294, 49401095274561608, 2402478324494963930, 119201977436336120482, 6017223412990713126034, 308361587173800754305214, 16013543997544827365960598
Offset: 0
G.f.: A(x) = 1 + x + 17*x^2 + 260*x^3 + 7244*x^4 + 214257*x^5 + 7593707*x^6 +...
log(A(x)) = x + 33*x^2/2 + 730*x^3/3 + 27425*x^4/4 + 1015626*x^5/5 + 43437282*x^6/6 + 1924149396*x^7/7 +...+ A069865(n)/2*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^6)/2*x^m/m)+x*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))
Showing 1-4 of 4 results.
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