A346061
A(n,k) = n! * [x^n] (Sum_{j=0..n} k^(j*(j+1)/2) * x^j/j!)^(1/k) if k>0, A(n,0) = 0^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 23, 1, 0, 1, 1, 13, 199, 393, 1, 0, 1, 1, 21, 901, 17713, 13729, 1, 0, 1, 1, 31, 2861, 249337, 4572529, 943227, 1, 0, 1, 1, 43, 7291, 1900521, 264273961, 3426693463, 126433847, 1, 0
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, ...
0, 1, 3, 7, 13, 21, ...
0, 1, 23, 199, 901, 2861, ...
0, 1, 393, 17713, 249337, 1900521, ...
0, 1, 13729, 4572529, 264273961, 6062674201, ...
...
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A:= (n, k)-> `if`(k>0, n!*coeff(series(add(k^(j*(j+1)/2)*
x^j/j!, j=0..n)^(1/k), x, n+1), x, n), k^n):
seq(seq(A(n, d-n), n=0..d), d=0..10);
A178319
E.g.f.: ( Sum_{n>=0} 3^(n*(n + 1)/2) * x^n/n! )^(1/3).
Original entry on oeis.org
1, 1, 7, 199, 17713, 4572529, 3426693463, 7575807034711, 49908659904426337, 983868034228748840161, 58130023275752925902247847, 10299771730830080877230000021479, 5474153833417147528343683843805979793, 8727821227226586439546709016484604992020049
Offset: 0
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 199*x^3/3! + 17713*x^4/4! +...
A(x)^3 = 1 + 3*x + 3^3*x^2/2! + 3^6*x^3/3! + 3^10*x^4/4! +...
Let E(x, q) = Sum_{n>=0} q^(n*(n - 1)/2)*x^n/n!, then the coefficients of (x^n/n!) in E(qx, q)^(1/q) begin:
1;
1;
q^2 - q + 1;
q^5 - 3*q^3 + 5*q^2 - 3*q + 1;
q^9 - 4*q^6 + q^5 + 15*q^4 - 24*q^3 + 17*q^2 - 6*q + 1;
q^14 - 5*q^10 + 5*q^9 - 10*q^8 + 30*q^7 - 95*q^5 + 149*q^4 - 110*q^3 + 45*q^2 - 10*q + 1; ...
Setting q = 3 generates this sequence.
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a:= n-> n!*coeff(series(add(3^binomial(j+1, 2)
*x^j/j!, j=0..n)^(1/3), x, n+1), x, n):
seq(a(n), n=0..14); # Alois P. Heinz, Mar 15 2021
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{a(n)=n!*polcoeff(sum(m=0,n,3^(m*(m+1)/2)*x^m/m!+x*O(x^n))^(1/3),n)}
A342578
a(n) = n! * [x^n] (Sum_{j>=0} n^(j*(j+1)/2) * x^j/j!)^(1/n) for n > 0, a(0) = 1.
Original entry on oeis.org
1, 1, 3, 199, 249337, 6062674201, 3653786369479951, 65709007885111803731947, 40564683796482484146182142025377, 969773549559254966290998252899999751714721, 999999990999996719397362087568018696141879478712251051, 49037072510879011742983689973641327840345400616866967292640434759551
Offset: 0
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a:= n-> `if`(n>0, coeff(series(add(n^binomial(j+1, 2)*
x^j/j!, j=0..n)^(1/n), x, n+1), x, n)*n!, 1):
seq(a(n), n=0..12);
Showing 1-3 of 3 results.
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