A208237
G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + k*x) / (1 + k*x + k^2*x^2).
Original entry on oeis.org
1, 1, 2, 5, 15, 54, 223, 1045, 5474, 31685, 200895, 1384470, 10304431, 82376101, 703949762, 6403761365, 61784985615, 630180031734, 6775001385343, 76572619018165, 907658144193314, 11259399965148005, 145879271404693215, 1970471655222795990, 27702625497930064591
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 223*x^6 + 1045*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!*x^2*(1+x)*(1+2*x)/((1+x+x^2)*(1+2*x+4*x^2)) + 3!*x^3*(1+x)*(1+2*x)*(1+3*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)) + 4!*x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)*(1+4*x+16*x^2)) +...
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{a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (1+k*x)/(1+k*x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))
A221371
O.g.f.: Sum_{n>=0} n!^2 * x^n * Product_{k=1..n} (1 + x) / (1 + k^2*x + k^2*x^2).
Original entry on oeis.org
1, 1, 4, 23, 209, 2744, 49539, 1180281, 35921892, 1360513711, 62770245601, 3466178083312, 225719029475675, 17117740162448105, 1495526385479298140, 149120758170390404103, 16831018302445533666705, 2134813624482300873515304, 302332062412598445891728563
Offset: 0
O.g.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 209*x^4 + 2744*x^5 + 49539*x^6 + ...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!^2*x^2*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)) + 3!^2*x^3*(1+x)*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)*(1+9*x+9*x^2)) + 4!^2*x^4*(1+x)*(1+x)*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)*(1+9*x+9*x^2)*(1+16*x+16*x^2)) + ...
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a[n1_Integer?NonNegative, n2_Integer?NonNegative] := CoefficientList[Sum[(m!)^2*x^m*Product[(1 + x)/(1 + k^2*x + k^2*x^2), {k, 1, m}], {m, 0, n2 + 1}] + O[x]^(n2 + 2), x][[n1 + 1 ;; n2 + 1]]; a[0, 18] (* Robert P. P. McKone, Sep 16 2023 *)
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{a(n)=polcoeff( sum(m=0, n, m!^2*x^m*prod(k=1, m, (1+x)/(1+k^2*x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))
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{a(n)=sum(k=0,n\2,binomial(n-k,k)*2*(-1)^(n-k+1)*(1-4^(n-k+1))*bernfrac(2*(n-k+1)))}
for(n=0, 30, print1(a(n), ", "))
A210438
G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k^2 + x) / (1 + k^2*x + x^2).
Original entry on oeis.org
1, 1, 4, 20, 174, 2262, 40894, 980590, 30095022, 1149982990, 53521460958, 2980006437662, 195562339712590, 14936971352026094, 1313606920832545022, 131776096083434471678, 14956389843996883667182, 1906794751364126563388238, 271321222225812454677233694
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 174*x^4 + 2262*x^5 + 40894*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+x)*(4+x)/((1+x+x^2)*(1+4*x+x^2)) + x^3*(1+x)*(4+x)*(9+x)/((1+x+x^2)*(1+4*x+x^2)*(1+9*x+x^2)) + x^4*(1+x)*(4+x)*(9+x)*(16+x)/((1+x+x^2)*(1+4*x+x^2)*(1+9*x+x^2)*(1+16*x+x^2)) +...
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{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (k^2+x)/(1+k^2*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
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