A250286
Number of permutations p of [n] such that p(i) > p(i+1) iff i == 0 (mod 9).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 54, 219, 714, 2001, 5004, 11439, 24309, 48619, 831384, 9069651, 64369341, 355150566, 1635163542, 6542615421, 23369110326, 75953123676, 227864057851, 5742168041637, 90830731860000, 920922875075934, 7159714782188364
Offset: 0
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
`if`(t=0, add(b(u-j, o+j-1, irem(t+1, 9)), j=1..u),
add(b(u+j-1, o-j, irem(t+1, 9)), j=1..o)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..35);
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nmax = 30; CoefficientList[Series[1 + Sum[(x^(9 - k) * HypergeometricPFQ[{1}, {10/9 - k/9, 11/9 - k/9, 4/3 - k/9, 13/9 - k/9, 14/9 - k/9, 5/3 - k/9, 16/9 - k/9, 17/9 - k/9, 2 - k/9}, -x^9/387420489])/(9 - k)!, {k, 0, 8}] / HypergeometricPFQ[{}, {1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9}, -x^9/387420489], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 21 2021 *)
A250287
Number of permutations p of [n] such that p(i) > p(i+1) iff i == 0 (mod 10).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 65, 285, 1000, 3002, 8007, 19447, 43757, 92377, 184755, 3527140, 42031760, 326057040, 1961245375, 9812764391, 42530831916, 164059546366, 574224816166, 1850302218766, 5550936701311, 156435448534980, 2711548312208295
Offset: 0
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
`if`(t=0, add(b(u-j, o+j-1, irem(t+1, 10)), j=1..u),
add(b(u+j-1, o-j, irem(t+1, 10)), j=1..o)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..35);
-
nmax = 30; CoefficientList[Series[1 + Sum[(x^(10 - k) * HypergeometricPFQ[{1}, {11/10 - k/10, 6/5 - k/10, 13/10 - k/10, 7/5 - k/10, 3/2 - k/10, 8/5 - k/10, 17/10 - k/10, 9/5 - k/10, 19/10 - k/10, 2 - k/10}, -x^10/10000000000])/(10 - k)!, {k, 0, 9}] / HypergeometricPFQ[{}, {1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10}, -x^10/10000000000], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 21 2021 *)
A181992
n-alternating permutations of length n^2.
Original entry on oeis.org
1, 1, 5, 1513, 60376809, 613498040952501, 2655748106132754540814283, 7350748555338515554166266981278924209, 18155845241010181420704703186769135339279915667193169, 53121946985233865823079732996510797894348260342024814486694637630897821
Offset: 0
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A181992 := proc(n) local E, dim, i, k; dim := n*n;
E := array(0..dim, 0..dim); E[0, 0] := 1;
for i from 1 to dim do
if i mod n = 0 then E[i, 0] := 0 ;
for k from i-1 by -1 to 0 do E[k, i-k] := E[k+1, i-k-1] + E[k, i-k-1] od;
else E[0, i] := 0;
for k from 1 by 1 to i do E[k, i-k] := E[k-1, i-k+1] + E[k-1, i-k] od;
fi od;
E[0, dim] end:
seq(A181992(i),i=0..9);
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A181985[n_, len_] := Module[{e, dim = n*(len - 1)}, e[0, 0] = 1; For[i = 1, i <= dim, i++, If[Mod[i, n] == 0, e[i, 0] = 0; For[k = i - 1, k >= 0, k--, e[k, i - k] = e[k + 1, i - k - 1] + e[k, i - k - 1]], e[0, i] = 0; For[k = 1, k <= i, k++, e[k, i - k] = e[k - 1, i - k + 1] + e[k - 1, i - k]]]]; Table[e[0, n*k], {k, 0, len - 1}]]; a[n_] := A181985[n, n + 1][[n + 1]]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 17 2013, after Maple code in A181985 *)
A164575
a(n) = n! * [x^n] 2*(tan(x))^2*(sec(x) + tan(x)).
Original entry on oeis.org
0, 0, 4, 12, 56, 240, 1324, 7392, 49136, 337920, 2652244, 21660672, 196658216, 1859020800, 19192151164, 206057828352, 2385488163296, 28669154426880, 367966308562084, 4893320282898432, 68978503204900376, 1005520890400604160, 15445185289163949004, 244890632417194278912
Offset: 0
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gf := (2*sin(x)*tan(x))/(1 - sin(x)): ser := series(gf, x, 25):
seq(n!*coeff(ser, x, n), n=0..23); # Peter Luschny, Aug 19 2019
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CoefficientList[Series[2Tan[x]^2(Sec[x]+Tan[x]),{x,0,23}],x]*Table[n!,{n,0,23}]
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my(x='x+O('x^30)); concat([0,0], Vec(serlaplace(2*(tan(x))^2*(1/cos(x) + tan(x))))) \\ Michel Marcus, Aug 13 2019
A309845
Expansion of e.g.f.: sec(x) + 2*tan(x).
Original entry on oeis.org
1, 2, 1, 4, 5, 32, 61, 544, 1385, 15872, 50521, 707584, 2702765, 44736512, 199360981, 3807514624, 19391512145, 419730685952, 2404879675441, 58177770225664, 370371188237525, 9902996106248192, 69348874393137901, 2030847773013704704, 15514534163557086905, 493842960380415967232
Offset: 0
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CoefficientList[Series[Sec[x]+2Tan[x],{x,0,25}],x]*Table[n!,{n,0,25}]
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my(x='x+O('x^30)); Vec(serlaplace(1/cos(x)+2*tan(x))) \\ Michel Marcus, Aug 20 2019
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