A004211 -id:A004211 - 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

A351143 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 0, 1, 2, 5, 16, 61, 258, 1177, 5776, 30537, 173394, 1050045, 6732608, 45459493, 322141106, 2390075249, 18525967328, 149684238801, 1257802518754, 10969260208565, 99100423076912, 926030783479629, 8937741026924450, 88988433270106249, 912906193294355952

Offset: 0

Ilya Gutkovskiy, Feb 02 2022

nonn

Shifts 2 places left under 2nd-order binomial transform.

Cf. A000994, A004211, A007472, A351144, A351150, A351151, A351152.

bintr:= proc(p) local b;
          b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
        end:
b:= (bintr@@2)(a):
a:= n-> `if`(n<2, 1-n, b(n-2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 07 2025

nmax = 25; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
(* another program *)
B[x_] := BesselK[1, 1]*BesselI[0, Exp[x]] + BesselI[1, 1]*BesselK[0, Exp[x]];
a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!
Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)

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

E.g.f.: BesselK(1, 1)*BesselI(0, exp(x)) + BesselI(1, 1)*BesselK(0, exp(x)). - Ven Popov, Apr 25 2025

A154602 Exponential Riordan array [exp(sinh(x)exp(x)), sinh(x)exp(x)].

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 49, 104, 70, 16, 1, 257, 641, 550, 190, 25, 1, 1539, 4380, 4531, 2080, 425, 36, 1, 10299, 32803, 39515, 22491, 6265, 833, 49, 1, 75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1, 609441, 2337505, 3575820, 2792476, 1192086, 281190, 36204, 2460, 81, 1

Offset: 0

Paul Barry, Jan 12 2009

Triangle T(n,k), read by rows, given by [1,2,1,4,1,6,1,8,1,10,1,12,1,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 02 2009

			Triangle begins
     1;
     1,    1;
     3,    4,    1;
    11,   19,    9,    1;
    49,  104,   70,   16,   1;
   257,  641,  550,  190,  25,  1;
  1539, 4380, 4531, 2080, 425, 36, 1;
Production matrix of this array is
  1,  1,
  2,  3,  1,
  0,  4,  5,  1,
  0,  0,  6,  7,  1,
  0,  0,  0,  8,  9,  1,
  0,  0,  0,  0, 10, 11,  1
with generating function exp(t*x)*(1+t)*(1+2*x).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Columns k=0..3 give A000007, A383203, A383204, A383205.
Cf. A004211 (first column), A256893.
Sums include: A000007 (alternating sign row), A055882 (row sums).

A154602:= func< n,k | (&+[2^(n-j)*Binomial(j,k)*StirlingSecond(n,j): j in [k..n]]) >;
[A154602(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 19 2024

A154602 := (n, k) -> add(2^(n-j) * binomial(j, k) * Stirling2(n, j), j = k..n): for n from 0 to 6 do seq(A154602(n, k), k = 0..n) od; # Peter Luschny, Dec 13 2022

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[Sinh[#] Exp[#]]&, Sinh[#] Exp[#]&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

def A154602(n,k): return sum(2^(n-j)*binomial(j,k)* stirling_number2(n,j) for j in range(k,n+1))
flatten([[A154602(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 19 2024

T(n, 0) = A004211(n).
Sum_{k=0..n} T(n, k) = A055882(n) (row sums).
From Peter Bala, Jun 15 2009: (Start)
T(n,k) = Sum_{i = k..n} 2^(n-i)*binomial(i,k)*Stirling2(n,i).
E.g.f.: exp((t+1)/2*(exp(2*x)-1)) = 1 + (1+t)*x + (3+4*t+t^2)*x^2/2! + ....
Row generating polynomials R_n(x):
R_n(x) = 2^n*Bell(n,(x+1)/2), where Bell(n,x) = Sum_{k = 0..n} Stirling2(n, k)*x^k denotes the n-th Bell polynomial.
Recursion:
R(n+1,x) = (x+1)*(R_n(x) + 2*d/dx(R_n(x))).
(End)
Recurrence: T(n,k) = 2*(k+1)*T(n-1,k+1) + (2*k+1)*T(n-1,k) + T(n-1,k-1). - Emanuele Munarini, Apr 14 2020
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n). - G. C. Greubel, Sep 19 2024
E.g.f. of column k (with leading zeros): f(x)^k * exp(f(x)) / k! with f(x) = (exp(2*x) - 1)/2. - Seiichi Manyama, Apr 19 2025

A241578 Square array read by antidiagonals upwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 15, 1, 1, 5, 19, 49, 52, 1, 1, 6, 29, 109, 257, 203, 1, 1, 7, 41, 201, 742, 1539, 877, 1, 1, 8, 55, 331, 1657, 5815, 10299, 4140, 1, 1, 9, 71, 505, 3176, 15821, 51193, 75905, 21147, 1, 1, 10, 89, 729, 5497, 35451, 170389, 498118, 609441, 115975, 1

Offset: 0

N. J. A. Sloane, Apr 29 2014

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...
1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, ...
1, 4, 19, 109, 742, 5815, 51193, 498118, 5296321, 60987817, 754940848, 9983845261, ...
1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, ...
1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, ...
1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, ...
1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, ...
...

Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

Three versions of this array are A111673, A241578, A241579.

Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

with(combinat):
T:=(n,k)->add(n^(k-j)*stirling2(k,j),j=1..k);
r:=n->[seq(T(n,k),k=1..12)];
for n from 0 to 8 do lprint(r(n)); od:

A337592 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 2^(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 4, 28, 312, 4936, 104128, 2806336, 93560064, 3765265408, 179415074304, 9964625629696, 636737424291840, 46303081167540224, 3796275000959266816, 348100339275620651008, 35448445862069986361344, 3984266642444252234153984, 491556877841462376382332928

Offset: 0

Ilya Gutkovskiy, Sep 02 2020

nonn

Cf. A004211, A337593, A337594, A337595, A337597.

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[2 x]] - 1)/2], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(2*x)) - 1) / 2).

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

A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

0, 1, 0, 1, 4, 13, 44, 173, 792, 4009, 21608, 122761, 737340, 4696341, 31665076, 224846037, 1672266352, 12976252561, 104816144656, 880061135057, 7670326372532, 69286959112797, 647568753568636, 6251768635591613, 62255057942504968, 638658964709824185

Offset: 0

Ilya Gutkovskiy, Feb 03 2022

nonn

Shifts 2 places left under 2nd-order binomial transform.

Cf. A000995, A004211, A007472, A351053, A351128, A351132, A351143, A351161.

bintr:= proc(p) local b;
          b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
        end:
b:= (bintr@@2)(a):
a:= n-> `if`(n<2, n, b(n-2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 07 2025

nmax = 25; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}];
(* another pprogram *)
B[x_] := BesselK[0, 1]*BesselI[0, Exp[x]] - BesselI[0, 1]*BesselK[0, Exp[x]];
a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!;
Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)

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

E.g.f.: BesselK(0, 1)*BesselI(0, exp(x)) - BesselI(0, 1)*BesselK(0, exp(x)). - Ven Popov, Apr 25 2025

A351756 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 2x)) / (1 - 2x)^2.

Original entry on oeis.org

1, 1, 5, 23, 119, 709, 4749, 35031, 281271, 2438565, 22673021, 224739303, 2363075191, 26246762213, 306830932749, 3763323446487, 48292462190743, 646763208308421, 9020009372203965, 130737162573013159, 1965798562640921879, 30613694640191725381

Offset: 0

Ilya Gutkovskiy, Feb 18 2022

nonn

Cf. A004211, A040027, A122704, A351757.

nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - 2 x)]/(1 - 2 x)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

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

A375173 Expansion of e.g.f. exp( (1/(1 - 4*x)^(1/2) - 1)/2 ).

Original entry on oeis.org

1, 1, 7, 79, 1225, 24121, 575311, 16105447, 517380529, 18752175505, 756760712311, 33645775575391, 1633792107752377, 86022043957561609, 4880923725657950335, 296882100064302393271, 19269430292162925519841, 1329278651404123963041697

Offset: 0

Seiichi Manyama, Aug 02 2024

For k >= 2, the difference a(n+k) - a(n) is divisible by k. It follows that for each k, the sequence formed by taking a(n) modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes [1, 1, 7, 9, 5, 1, 1, 7, 9, 5, ...], a purely periodic sequence of period 5. Cf. A047974. - Peter Bala, Feb 11 2025

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A080893, A375175.

Cf. A004211.

Table[4^n * Sum[Abs[StirlingS1[n, k]] * BellB[k, 1/2] / 2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 02 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-4*x)^(1/2)-1)/2)))

a(n) = Sum_{k=0..n} 4^(n-k) * |Stirling1(n,k)| * A004211(k) = 4^n * Sum_{k=0..n} (1/2)^k * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
From Vaclav Kotesovec, Aug 02 2024: (Start)
a(n) = 6*(2*n - 3)*a(n-1) - (48*n^2 - 192*n + 191)*a(n-2) + 32*(n-3)*(n-2)*(2*n - 5)*a(n-3).
a(n) ~ 2^(2*n - 1/6) * n^(n - 1/3) / (sqrt(3) * exp(n - 3*2^(-4/3)*n^(1/3) + 1/2)) * (1 - 31/(72*2^(2/3)*n^(1/3)) - 4607/(20736*2^(1/3)*n^(2/3))). (End)
a(n) = (1/exp(1/2)) * (-4)^n * n! * Sum_{k>=0} binomial(-k/2,n)/(2^k * k!). - Seiichi Manyama, Jan 18 2025

A059340 Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 15, 23, 10, 1, 52, 109, 76, 19, 1, 203, 544, 531, 224, 36, 1, 877, 2876, 3641, 2204, 631, 69, 1, 4140, 16113, 25208, 20089, 8471, 1749, 134, 1, 21147, 95495, 178564, 177631, 100171, 31331, 4838, 263, 1

Offset: 1

Vladeta Jovovic, Jan 27 2001

Essentially triangle given by [1,1,1,2,1,3,1,4,1,5,1,6,...] DELTA [0,1,0,2,0,3,0,4,0,5,0,6,...] = [1;1,0;2,1,0;5,5,1,0;15,23,10,1,0;...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 20 2006

			Triangle starts:
   1;
   2,   1;
   5,   5,   1;
  15,  23,  10,   1;
  52, 109,  76,  19,   1;

G. C. Greubel, Table of n, a(n) for n = 1..1275

Row sums = A004211, T(n,0) = A000110.

Table[Sum[StirlingS2[n, j]*Binomial[n - j, k], {j, 0, n}], {n, 1,
  5}, {k, 0, n - 1}] (* G. C. Greubel, Jan 07 2017 *)

T = lambda n,k: sum(stirling_number2(n,j)*binomial(n-j,k) for j in (0..n))
# Also "for n in (0..11): print([T(n,k) for k in (0..n)])" makes sense.
for n in (1..11): print([T(n,k) for k in (0..n-1)]) # Peter Luschny, Aug 06 2015

T(n,k) = Sum_{i=0..n} stirling2(n, n-i)*binomial(i, k).

T(n,k) = Sum_{i=0..n} stirling2(n, i)*binomial(n-i, k). - Peter Luschny, Aug 06 2015

A241577 n^3 + 4n^2 - 5n + 1.

Original entry on oeis.org

1, 1, 15, 49, 109, 201, 331, 505, 729, 1009, 1351, 1761, 2245, 2809, 3459, 4201, 5041, 5985, 7039, 8209, 9501, 10921, 12475, 14169, 16009, 18001, 20151, 22465, 24949, 27609, 30451, 33481, 36705, 40129, 43759, 47601, 51661, 55945, 60459, 65209, 70201, 75441, 80935, 86689, 92709, 99001, 105571, 112425, 119569

Offset: 0

N. J. A. Sloane, Apr 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. See Eq. (7), col. 4.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[n^3+4*n^2-5*n+1: n in [0..50]]; // Vincenzo Librandi, Apr 28 2014

CoefficientList[Series[(1 - 3 x + 17 x^2 - 9 x^3)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 28 2014 *)

a(n)=n^3+4*n^2-5*n+1 \\ Charles R Greathouse IV, Aug 26 2014

G.f.: (1-3*x+17*x^2-9*x^3)/(1-x)^4. - Vincenzo Librandi, Apr 28 2014

Recurrence: a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Fung Lam, May 11 2014

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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A351143 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).

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A154602 Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].

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A241578 Square array read by antidiagonals upwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

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A337592 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 2^(k-1) * a(n-k).

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A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).

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Maple

Mathematica

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A351756 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 2*x)) / (1 - 2*x)^2.

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Mathematica

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A375173 Expansion of e.g.f. exp( (1/(1 - 4*x)^(1/2) - 1)/2 ).

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A351143 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2x)) / (1 - 2x).

A154602 Exponential Riordan array [exp(sinh(x)exp(x)), sinh(x)exp(x)].

A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2x)) / (1 - 2x).

A351756 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 2x)) / (1 - 2x)^2.

A241577 n^3 + 4n^2 - 5n + 1.