A361616
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 15, 34, 1, 1, 5, 25, 103, 209, 1, 1, 6, 37, 214, 885, 1546, 1, 1, 7, 51, 373, 2293, 9051, 13327, 1, 1, 8, 67, 586, 4721, 29176, 106843, 130922, 1, 1, 9, 85, 859, 8481, 70981, 427189, 1425495, 1441729, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
1, 7, 15, 25, 37, 51, ...
1, 34, 103, 214, 373, 586, ...
1, 209, 885, 2293, 4721, 8481, ...
1, 1546 ,9051, 29176, 70981, 146046, ...
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T(n,k) = n! * sum(j=0, n, binomial(n+(k-1)*(j+1), n-j)/j!);
A361626
Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x)^2.
Original entry on oeis.org
1, 3, 17, 139, 1437, 17711, 252133, 4059567, 72779129, 1435276027, 30836352441, 716101686323, 17858449006357, 475653606922599, 13467411746316557, 403708230041927191, 12767545998797849073, 424670548932688771187, 14814998283177691422049
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)^2))
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a(n)=n! * sum(k=0, n, binomial(n+2*k+1,n-k)/k!) \\ Winston de Greef, Mar 18 2023
A375225
Expansion of e.g.f. exp( x^2/(1-x)^3 ) / (1-x)^3.
Original entry on oeis.org
1, 3, 14, 96, 876, 9780, 127200, 1877400, 30947280, 563114160, 11202135840, 241655641920, 5614182826560, 139647350082240, 3700648372861440, 104032358410780800, 3090961262246457600, 96747013002684844800, 3180863885996673676800, 109570715078766355814400
Offset: 0
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With[{nn=20},CoefficientList[Series[Exp[x^2/(1-x)^3]/(1-x)^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 20 2024 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x^2/(1-x)^3)/(1-x)^3))
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a(n) = n!*sum(k=0, n\2, binomial(n+2+k, n-2*k)/k!);
A377967
Expansion of e.g.f. (1+x)^3 * exp(x*(1+x)^3).
Original entry on oeis.org
1, 4, 19, 124, 961, 8236, 79339, 840484, 9595009, 117764596, 1542837091, 21406165804, 313381177729, 4822681240924, 77704955681851, 1307128152596116, 22899018541506049, 416756647023727204, 7863586717014612019, 153550319029835965276, 3097694623619639050561
Offset: 0
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With[{nn=20},CoefficientList[Series[(1+x)^3 Exp[x*(1+x)^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2025 *)
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a(n, s=3, t=3) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);
Showing 1-4 of 4 results.