A345969
Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x))).
Original entry on oeis.org
1, 0, 1, 2, 18, 104, 1015, 9666, 116557, 1504856, 22300704, 358916480, 6373675825, 122332173300, 2540560235161, 56558354414870, 1346402030278050, 34093192112537888, 915570658175517151, 25983157665663651150, 777141557158947654637, 24430880483991543481580
Offset: 0
1/sqrt(3-2/((1-x)*exp(x))) = 1 + x^2/2! + 2*x^3/3! + 18*x^4/4! + 104*x^5/5! + 1015*x^6/6! + 9666*x^7/7! + 116557*x^8/8! + 1504856*x^9/9! + ...
a(17) = Sum_{k=1..8} A305404(k)*A008306(17,k) = 34093192112537888.
For k=1, A305404(1)*A008306(17,1) == -1 (mod 17), because A305404(1) = 1 and A008306(17,1) = (17-1)!
For k>=2, A305404(k)*A008306(17,k) == 0 (mod 17), because A008306(17,k) == 0 (mod 17), result a(17) == -1 (mod 17).
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A305404:= n-> add(Stirling2(n,k)*doublefactorial(2*k-1), k=0..n):
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a := n-> add((A305404(k)*A008306(n, k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(1/sqrt(3-2/((1-x)*exp(x))), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
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CoefficientList[Series[1/Sqrt[3-2/((1-x)*E^x)], {x, 0, 24}], x] * Range[0, 24]!
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my(x='x+O('x^30)); Vec(serlaplace(1/sqrt(3 - 2 / ((1 - x)*exp(x))))) \\ Michel Marcus, Jul 01 2021
A305405
Expansion of Sum_{k>=0} k!!*x^k/Product_{j=1..k} (1 - j*x).
Original entry on oeis.org
1, 1, 3, 10, 41, 201, 1126, 7043, 48603, 366298, 2987189, 26163501, 244654150, 2430411335, 25539609327, 282834656434, 3290175964577, 40089424302657, 510340938343270, 6772086558823547, 93481666812344979, 1339885322519303434, 19907413622297965373, 306126204811557339045
Offset: 0
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b:= proc(n, m) option remember;
`if`(n=0, doublefactorial(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..23); # Alois P. Heinz, Aug 04 2021
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nmax = 23; CoefficientList[Series[Sum[k!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1 + Exp[(E^x - 1)^2/2] (Exp[x] - 1) (1 + Sqrt[Pi/2] Erf[(Exp[x] - 1)/Sqrt[2]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] k!!, {k, 0, n}], {n, 0, 23}]
A375695
Expansion of e.g.f. 1 / sqrt(1 - 2 * x * (exp(x) - 1)).
Original entry on oeis.org
1, 0, 2, 3, 40, 185, 2436, 20797, 307616, 3869217, 66259900, 1091351261, 21671302368, 437191547377, 9981020325836, 236821065758565, 6144729994822336, 167019469703969345, 4868403452056231164, 148845363155530699789, 4822574537456548631360
Offset: 0
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With[{nn=20},CoefficientList[Series[1/Sqrt[1-2x (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 06 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-2*x*(exp(x)-1))))
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a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = n!*sum(k=0, n, a001147(k)*stirling(n-k, k, 2)/(n-k)!);
A375991
Expansion of e.g.f. (3 - 2 * exp(x))^(3/2).
Original entry on oeis.org
1, -3, 0, 9, 45, 252, 1935, 19989, 260190, 4063887, 73823445, 1527002694, 35408499885, 909389617497, 25618701424680, 785355764569749, 26024092206299505, 926859918577582332, 35306305954587340515, 1432301360556686816529, 61649353087003554947550
Offset: 0
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With[{nn=20},CoefficientList[Series[(3-2Exp[x])^(3/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2025 *)
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a(n) = sum(k=0, n, prod(j=0, k-1, 2*j-3)*stirling(n, k, 2));
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