A007405 -id:A007405 - cp's OEIS frontend

A187251 Number of permutations of [n] having no cycle with 3 or more alternating runs (it is assumed that the smallest element of a cycle is in the first position).

1, 1, 2, 6, 22, 94, 460, 2532, 15420, 102620, 739512, 5729192, 47429896, 417429800, 3888426512, 38192416048, 394239339792, 4264424937488, 48212317486112, 568395755184224, 6973300915138656, 88860103591344864, 1174131206436335296, 16061756166912244800

Offset: 0

Emeric Deutsch, Mar 08 2011

nonn

a(n) = A187250(n,0).
It appears that a(n) = A216964(n,1), for n>0. - Michel Marcus, May 17 2013.
The above comment is correct. Let b(n) be the n-th element of the first column of the triangle in A216964. By definition, b(n) is the number of permutations of [n] with no cyclic valleys. Recall that alternating runs of permutations are monotonically increasing or decreasing subsequences. In other words, b(n) is the number of permutations of [n] with the restriction that every cycle has at most two alternating runs, so b(n) = A187251(n) = a(n). - Shi-Mei Ma, May 18 2013.

			a(4)=22 because only the permutations 3421=(1324) and 4312=(1423) have cycles with more than 2 alternating runs.

Alois P. Heinz, Table of n, a(n) for n = 0..528
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 56.

Cf. A001519, A187245, A187248, A187250, A216964.

Row sums of A344855.

g := exp((2*z-1+exp(2*z))*1/4): gser := series(g, z = 0, 28): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 23);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*ceil(2^(j-2)), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 30 2021

nmax = 20; CoefficientList[Series[E^((2*x-1+E^(2*x))/4), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 17 2020 *)

a(n):=n!*sum(2^(n-2*k)*sum(binomial(k,j)*stirling2(n-k+j,j)*j!/(n-k+j)!,j,0,k)/k!,k,1,n); /* Vladimir Kruchinin, Apr 25 2011 */

x='x+O('x^66); Vec(serlaplace(exp( (2*x-1+exp(2*x))/4 ))) /* Joerg Arndt, Apr 26 2011 */

lista(m) = {P = x*y; for (n=1, m, M = subst(P, x, 1); M = subst(M, y, 1); print1(polcoeff(M, 0, q), ", "); P = (n*q+x*y)*P + 2*q*(1-q)*deriv(P, q)+ 2*x*(1-q)*deriv(P, x)+ (1-2*y+q*y)*deriv(P, y); ); } \\ (adapted from PARI prog in A216964) \\ Michel Marcus, May 17 2013

E.g.f.: exp( (2*z-1+exp(2*z))/4 ).
For n>=1: a(n)=n!*sum(k=1..n, 2^(n-2*k)*sum(j=0..k, binomial(k,j)*stirling2(n-k+j,j)*j!/(n-k+j)!)/k!); [From Vladimir Kruchinin, Apr 25 2011]
G.f.: 1/Q(0) where Q(k) =  1 - x*k - x/(1 - x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+1) - m*x^2*(k+1)/Q(k+1) and m=1 (continued fraction); setting m=2 gives A004211, m=4 gives A124311 without signs. - Sergei N. Gladkovskii, Sep 26 2013
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
Sum_{k=0..n} binomial(n,k) * a(k) * a(n-k) = A007405(n). - Vaclav Kotesovec, Apr 17 2020
a(n) = Sum_{j=1..n} a(n-j)*binomial(n-1,j-1)*ceiling(2^(j-2)) for n > 0, a(0) = 1. - Alois P. Heinz, May 30 2021

A355110 Expansion of e.g.f. 2 / (3 - 2x - exp(2x)).

Original entry on oeis.org

1, 2, 10, 76, 768, 9696, 146896, 2596448, 52449536, 1191944704, 30097334784, 835973778432, 25330620762112, 831497823494144, 29394162040580096, 1113330929935101952, 44979662118902366208, 1930798895281527717888, 87756941394038739828736, 4210241529540625311727616

Offset: 0

Ilya Gutkovskiy, Jun 19 2022

nonn

Cf. A004123, A006155, A007405, A122704, A343672, A355111, A355112, A355113, A355114.

nmax = 19; CoefficientList[Series[2/(3 - 2 x - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 2^(k-1) * a(n-k).

a(n) ~ n! / ((1 + LambertW(exp(3))) * ((3 - LambertW(exp(3)))/2)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (nk + 1)^n / (n^k k!).

Original entry on oeis.org

1, 2, 6, 35, 352, 5307, 111592, 3117900, 111259904, 4912490375, 261954304224, 16560019685937, 1222893826048000, 104189533522270666, 10132262911996769408, 1114216450970154278543, 137427598621356912082944, 18877351974681584403701519, 2869969478954093766868948480

Offset: 0

Ilya Gutkovskiy, Apr 16 2020

nonn

Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Cf. A000110, A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582, A301419, A334165.

Table[SeriesCoefficient[1/(1 - x) Sum[(x/(1 - x))^k/Product[(1 - n j x/(1 - x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! SeriesCoefficient[Exp[x + (Exp[n x] - 1)/n], {x, 0, n}], {n, 1, 18}]]

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^k / Product_{j=1..k} (1 - n*j*x/(1 - x)).
a(n) = n! * [x^n] exp(x + (exp(n*x) - 1) / n), for n > 0.
a(n) = A334165(n,n).

A364069 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b = 63.

Original entry on oeis.org

1, 2, 67, 4355, 295234, 21036803, 1625419909, 140823067772, 13947448935109, 1570142163116087, 196457384808738412, 26717651072732512841, 3896182904620308595021, 605803757139146097600266, 100236348400243756326661039, 17619174544126256877550593743, 3280792242500933388439611444802

Offset: 0

Stefano Spezia, Jul 04 2023

nonn

a(n) is the number of all 64-subgroups of R^n, where R^n is a near-vector space over a proper nearfield R.

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10), A364070 (b=624).
Row sums of the triangle A364072.
2nd row of the array A364074.

With[{m=16, b=63}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b], {x, 0, m}], x]*Range[0, m]!] (* or *)
a[n_]:=Sum[Sum[Binomial[n,d]StirlingS2[n-d,k]63^(n-d-k),{d,0,n-k}],{k,0,n}]; Array[a,17,0]

E.g.f.: exp(x + (exp(63*x) - 1)/63).
a(n) = exp(-1/63) * Sum_{k>=0} (63*k + 1)^n / (63^k * k!).
a(n) ~ 63^(n + 1/63) * n^(n + 1/63) * exp(n/LambertW(63*n) - n - 1/63) / (sqrt(1 + LambertW(63*n)) * LambertW(63*n)^(n + 1/63)).
a(n) = Sum_{k=0..n} Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*63^(n-d-k).

A364070 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=624.

Original entry on oeis.org

1, 2, 628, 393128, 247268752, 156500388128, 100264147266880, 65739252669562496, 44949841635462426880, 32961816599696140935680, 26763226019573589904012288, 24577197816669853786615064576, 25455086256328481246829666144256, 29063231104986184254344094194278400

Offset: 0

Stefano Spezia, Jul 04 2023

nonn

a(n) is the number of all 625-subgroups of R^n, where R^n is a near-vector space over a proper nearfield R.

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-8.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10), A364069 (b=63).
Row sums of the triangle A364073.
3rd row of the array A364074.

With[{m=13, b=624}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b], {x, 0, m}], x]*Range[0, m]!] (* or *)
a[n_]:=Sum[Sum[Binomial[n,d]StirlingS2[n-d,k]624^(n-d-k),{d,0,n-k}],{k,0,n}]; Array[a,14,0]

E.g.f.: exp(x + (exp(624*x) - 1)/624).
a(n) = exp(-1/624) * Sum_{k>=0} (624*k + 1)^n / (624^k * k!).
a(n) ~ 624^(n + 1/624) * n^(n + 1/624) * exp(n/LambertW(624*n) - n - 1/624) / (sqrt(1 + LambertW(624*n)) * LambertW(624*n)^(n + 1/624)).
a(n) = Sum_{k=0..n} Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*624^(n-d-k).

A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)k+1)T(n-1,k) in column c.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 5, 2, 1, 1, 16, 15, 6, 2, 1, 1, 32, 52, 24, 7, 2, 1, 1, 64, 203, 116, 35, 8, 2, 1, 1, 128, 877, 648, 214, 48, 9, 2, 1, 1, 256, 4140, 4088, 1523, 352, 63, 10, 2, 1, 1, 512, 21147, 28640, 12349, 3008, 536, 80, 11, 2, 1

Offset: 0

Gary W. Adamson, Aug 07 2005

Triangles of generalized Stirling numbers of the second kind may be defined by recurrences T(n,k) = T(n-1,k-1) + Q*T(n-1,k) initialized by T(0,0)=T(1,0)=T(1,1)=1. Q=1 generates Pascal's triangle A007318,
Q=k+1 generates A008277, Q=2k+1 generates A039755, Q=3k+1 generates A111577, Q=4k+1 generates A111578, Q=5k+1 generates A166973.
(These definitions assume row and column enumeration 0<=n, 0<=k<=n.)
Each of these triangles characterized by Q=(c-1)*k+1 has row sums sum_{k=0..n} T(n,k), which define the column A(.,c).

Cf. A008277, A000110, A039755, A004211, A111577, A111578.

T := proc(n,k,c) if k < 0 or k > n then 0 ; elif n <= 1 then 1; else procname(n-1,k-1,c)+((c-1)*k+1)*procname(n-1,k,c) ; fi; end:
A111579 := proc(r,c) local n; if c = 0 then 1 ; else n := r-c ; add( T(n,k,c),k=0..n) ; end if; end:
seq(seq(A111579(r,c),c=0..r),r=0..10) ; # R. J. Mathar, Oct 30 2009

T[n_, k_, c_] := T[n, k, c] = If[k < 0 || k > n, 0, If[n <= 1, 1, T[n-1, k-1, c] + ((c-1)*k+1)*T[n-1, k, c]]];
A111579[r_, c_] := Module[{n}, If[c == 0, 1, n = r - c; Sum[T[n, k, c], {k, 0, n}]]];
Table[A111579[r, c], {r, 0, 10}, {c, 0, r}] // Flatten (* Jean-François Alcover, Aug 01 2023, after R. J. Mathar *)

A(r=n+c,c) = sum_{k=0..n} T(n,k,c), 0<=c<=r where T(n,k,c) = T(n-1,k-1,c) + ((c-1)*k+1)*T(n-1,k,c).
A(r,0) = 1.
A(r,1) = 2^(r-1).
A(r,2) = A000110(r-1).
A(r,3) = A007405(r-3).

Edited by R. J. Mathar, Oct 30 2009

A334190 a(n) = exp(1/2) * Sum_{k>=0} (2k + 1)^n / ((-2)^k k!).

Original entry on oeis.org

1, 0, -2, -4, 4, 64, 248, 48, -6512, -51200, -171296, 830400, 17870400, 144684032, 441316224, -5976726784, -119879356160, -1123892297728, -3962230563328, 70410917051392, 1686366492509184, 19578100126072832, 101728414306826240, -1258662784047370240, -42727186269262737408

Offset: 0

Ilya Gutkovskiy, Apr 18 2020

sign

Column k=2 of A334192.

Cf. A007405, A009235, A293037, A308536, A334191.

nmax = 24; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 2 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[x + (1 - Exp[2 x])/2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k, -1/2], {k, 0, n}], {n, 0, 24}] (* Vaclav Kotesovec, Apr 18 2020 *)

G.f.: (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - 2*j*x/(1 - x)).

E.g.f.: exp(x + (1 - exp(2*x)) / 2).

A337011 a(n) = 2^n * exp(-1/2) * Sum_{k>=0} (k + 2)^n / (2^k * k!).

Original entry on oeis.org

1, 5, 27, 159, 1025, 7221, 55307, 457631, 4065569, 38566021, 388757083, 4146851583, 46636281185, 551163837685, 6825500514059, 88341860285631, 1192267628956353, 16743728349797765, 244221140242647579, 3693367920926321375, 57821628101627115329

Offset: 0

Ilya Gutkovskiy, Aug 11 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..513

Cf. A004211, A007405, A045379, A337010.

E:= exp(4*x+exp(2*x)/2-1/2):
S:= series(E,x,31):
seq(coeff(S,x,n)*n!,n=0..30); # Robert Israel, Aug 14 2020

nmax = 20; CoefficientList[Series[Exp[4 x + (Exp[2 x] - 1)/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 5 a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k - 1) a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 20}]

E.g.f.: exp(4*x + (exp(2*x) - 1) / 2).
a(0) = 1; a(n) = 5 * a(n-1) + Sum_{k=2..n} binomial(n-1,k-1) * 2^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 4^(n-k) * A004211(k).

A216373 G.f.: Sum_{n>=0} x^n / Product_{k=0..n} (1 - (2k-1)x)^2.

Original entry on oeis.org

1, 1, 3, 12, 65, 419, 3088, 25557, 233687, 2331092, 25130877, 290632455, 3583432896, 46864388137, 647273948043, 9406216355420, 143356121222905, 2284850518224363, 37988158312023376, 657378186247162493, 11816449728615690079, 220230214060016856164

Offset: 0

Paul D. Hanna, Sep 05 2012

nonn

Compare to o.g.f. of Dowling numbers: Sum_{n>=0} x^n / Product_{k=0..n} (1 - (2*k-1)*x).

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 65*x^4 + 419*x^5 + 3088*x^6 +...
where
A(x) = 1 + x/(1-x)^2 + x^2/((1-x)*(1-3*x))^2 + x^3/((1-x)*(1-3*x)*(1-5*x))^2 + x^4/((1-x)*(1-3*x)*(1-5*x)*(1-7*x))^2 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A216367, A007405.

{a(n)=polcoeff(sum(m=0, n, x^m/prod(k=1, m, 1-(2*k-1)*x +x*O(x^n))^2), n)}
for(n=0,30,print1(a(n),", "))

A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (kj + 1)^n / (k^j j!).

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 1, 2, 6, 15, 1, 2, 7, 24, 52, 1, 2, 8, 35, 116, 203, 1, 2, 9, 48, 214, 648, 877, 1, 2, 10, 63, 352, 1523, 4088, 4140, 1, 2, 11, 80, 536, 3008, 12349, 28640, 21147, 1, 2, 12, 99, 772, 5307, 29440, 112052, 219920, 115975, 1, 2, 13, 120, 1066, 8648, 60389, 324096, 1120849, 1832224, 678570

Offset: 0

Ilya Gutkovskiy, Apr 17 2020

Square array of Dowling numbers.

			Square array begins:
    1,    1,     1,     1,     1,     1,  ...
    2,    2,     2,     2,     2,     2,  ...
    5,    6,     7,     8,     9,    10,  ...
   15,   24,    35,    48,    63,    80,  ...
   52,  116,   214,   352,   536,   772,  ...
  203,  648,  1523,  3008,  5307,  8648,  ...

Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Columns k=1..10 give A000110 (for n > 0), A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582.

Cf. A241578, A241579, A334162 (diagonal).

Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[x + (Exp[k x] - 1)/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten

G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).

E.g.f. of column k: exp(x + (exp(k*x) - 1) / k).

cp's OEIS Frontend

A187251 Number of permutations of [n] having no cycle with 3 or more alternating runs (it is assumed that the smallest element of a cycle is in the first position).

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A355110 Expansion of e.g.f. 2 / (3 - 2*x - exp(2*x)).

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A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (n*k + 1)^n / (n^k * k!).

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A364069 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b = 63.

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A364070 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=624.

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A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)*k+1)*T(n-1,k) in column c.

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A334190 a(n) = exp(1/2) * Sum_{k>=0} (2*k + 1)^n / ((-2)^k * k!).

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A337011 a(n) = 2^n * exp(-1/2) * Sum_{k>=0} (k + 2)^n / (2^k * k!).

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A355110 Expansion of e.g.f. 2 / (3 - 2x - exp(2x)).

A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (nk + 1)^n / (n^k k!).

A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)k+1)T(n-1,k) in column c.

A334190 a(n) = exp(1/2) * Sum_{k>=0} (2k + 1)^n / ((-2)^k k!).

A216373 G.f.: Sum_{n>=0} x^n / Product_{k=0..n} (1 - (2k-1)x)^2.

A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (kj + 1)^n / (k^j j!).