A345652
Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).
Original entry on oeis.org
1, 0, -1, 2, 0, -16, 65, -78, -749, 6232, -22068, -28920, 1004685, -7408740, 22263215, 157632230, -2874256740, 21590948480, -53087332675, -956539294506, 16344490525835, -132605481091060, 294656170409328, 9113173803517344, -167298122286332823
Offset: 0
exp(-1+(x+1)*exp(-x)) = 1 - x^2/2! + 2*x^3/3! - 16*x^5/5! + 65*x^6/6! - 78*x^7/7! - 749*x^8/8! + 6232*x^9/9! + ...
Cf.
A292935 (without 1+x: EGF e^(e^(-x)-1)),
A000110 (absolute values: Bell numbers, EGF e^(e^x-1))
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a := series(exp(-1+(x+1)*exp(-x)), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);
a := proc(n) option remember; `if`(n=0, 1, add((n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k), k=0..n-2)) end: seq(a(n), n=0..24);
# third program:
A345652 := n -> add((-1)^(n-k)*combinat[bell](k)*A106828(n, k), k=0..iquo(n, 2)):
seq(A345652(n), n=0..24); # Mélika Tebni, Sep 21 2021
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nmax = 24; CoefficientList[Series[Exp[-1+(x+1)*Exp[-x]], {x, 0, nmax}], x] Range[0, nmax]!
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seq(n) = {Vec(serlaplace(exp(-1+(x+1)*exp(-x + O(x*x^n)))))} \\ Andrew Howroyd, Jun 21 2021
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a(n) = if(n==0, 1, sum(k=2, n, (-1)^(k-1)*(k-1)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Mar 15 2022
A348208
a(n) = Sum_{k=0..floor(n/2)} (-1)^(k-1)*(k-1)^2*A106828(n, k).
Original entry on oeis.org
-1, 0, 0, 0, -3, -20, -70, -84, 1267, 18824, 209484, 2284920, 26010369, 314864628, 4073158102, 56304102596, 830061867975, 13016975343184, 216535182535928, 3810394068301296, 70744547160678501, 1382375535029293500, 28364229790262962386, 609820072529413714012
Offset: 0
E.g.f.: -1 - 3*x^4/4! - 20*x^5/5! - 70*x^6/6! - 84*x^7/7! + 1267*x^8/8! + 18824*x^9/9! + ...
a(11) = Sum_{k=0..5} (-1)^(k-1)*(k-1)^2*A106828(11, k).
a(11) = (-1)*1*0 + (1)*0*3628800 + (-1)*1*6636960 + (1)*4*3678840 + (-1)*9*705320 + (1)*16*34650 = 2284920.
For k = 0, A106828(11,0) = 0.
For k = 1, (1-1)^2 = 0.
For 2 <= k <= 5, A106828(11, k) == 0 (mod 11*10).
Result a(11) == 0 (mod 11*10).
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a := series((-1+2*x-2*x^2+x^3+(1-x)*(log((1-x)^(1-2*x))-(log(1-x))^2))*exp(x), x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
# second program:
a := n -> add((-1)^(k-1)*(k-1)^2*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
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CoefficientList[Series[(-1+2*x-2*x^2+x^3+(1-x)*(Log[(1-x)^(1-2*x)]-(Log[1-x])^2))*Exp[x], {x, 0, 23}], x]*Range[0, 23]!
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my(x='x+O('x^30)); Vec(serlaplace((-1 + 2*x - 2*x^2 + x^3 + (1 - x)*(log((1 - x)^(1 - 2*x)) - (log(1 - x))^2))*exp(x))) \\ Michel Marcus, Oct 07 2021
A349959
a(n) = Sum_{k=0..floor(n/2)} (k-1)^2*A106828(n, k).
Original entry on oeis.org
1, 0, 0, 0, 3, 20, 190, 1764, 17773, 192632, 2250036, 28254600, 380304639, 5468906508, 83750505826, 1361579283596, 23431400945145, 425669127018416, 8142731710207432, 163636478165355408, 3447201944202849819, 75973975479088955460, 1748531872985454054246, 41951755708613404583732
Offset: 0
E.g.f.: 1 + 3*x^4/4! + 20*x^5/5! + 190*x^6/6! + 1764*x^7/7! + 17773*x^8/8! + 192632*x^9/9! + ...
a(13) = Sum_{k=0..6} (k-1)^2*A106828(13, k).
a(13) = 1*0 + 0*479001600 + 1*967524480 + 4*647536032 + 9*177331440 + 16*18858840 + 25*540540 = 5468906508.
For k = 0, A106828(13, 0) = 0.
For k = 1, (1-1)^2 = 0.
For 2 <= k <= 6, A106828(13, k) == 0 (mod 13*12).
Result a(13) == 0 (mod 13*12).
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a := n -> add((k-1)^2*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
# second program:
a := series((-2-x+(3+log((1-x)^(1+2*x))+(log(1-x))^2)/(1-x))/exp(x), x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
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CoefficientList[Series[(-2-x+(3+Log[(1-x)^(1+2*x)]+(Log[1-x])^2)/(1-x))/Exp[x], {x, 0, 23}], x]*Range[0, 23]!
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E2(n, m) = sum(k=0, n-m, (-1)^(n+k)*binomial(2*n+1, k)*stirling(2*n-m-k+1, n-m-k+1, 1)); \\ A008517
ast1(n, k) = if ((n==0) && (k==0), 1, sum(j=0, n-k, binomial(j, n-2*k)*E2(n-k, j+1))); \\ A106828
a(n) = sum(k=0, n\2, (k-1)^2*ast1(n, k)); \\ Michel Marcus, Dec 07 2021
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