A004211 -id:A004211 - cp's OEIS frontend

A241579 Square array read by antidiagonals downwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

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

A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. 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:

A261275 Number of set partitions C_t(n) of {1,2,...,t} into at most n parts, with an even number of elements in each part distinguished by marks; triangle C_t(n), t>=0, 0<=n<=t, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 4, 10, 11, 0, 8, 36, 48, 49, 0, 16, 136, 236, 256, 257, 0, 32, 528, 1248, 1508, 1538, 1539, 0, 64, 2080, 6896, 9696, 10256, 10298, 10299, 0, 128, 8256, 39168, 66384, 74784, 75848, 75904, 75905, 0, 256, 32896, 226496, 475136, 586352, 607520, 609368, 609440, 609441

Offset: 0

Mark Wildon, Aug 13 2015

C_t(n) is the number of sequences of t top-to-random shuffles that leave a deck of n cards invariant, if each shuffle is permitted to flip the orientation of the card it moves.
C_t(n) =  where pi is the permutation character of the hyperoctahedral group BSym_n = C_2 wreath Sym_n given by its imprimitive action on a set of size 2n. This gives a combinatorial interpretation of C_t(n) using sequences of box moves on pairs of Young diagrams.
C_t(t) is the number of set partitions of a set of size t with an even number of elements in each part distinguished by marks.
C_t(n) = C_t(t) if n > t.

			Triangle starts:
  1;
  0,  1;
  0,  2,    3;
  0,  4,   10,   11;
  0,  8,   36,   48,   49;
  0, 16,  136,  236,  256,   257;
  0, 32,  528, 1248, 1508,  1538,  1539;
  0, 64, 2080, 6896, 9696, 10256, 10298, 10299;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D, arXiv:1507.04803 [math.CO], 2015.

Columns n=0,1,2,3 give A000007, A000079, A007582, A233162 (proved for n=3 in reference above).
Main diagonal gives A004211.
Cf. A075497.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
       `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
        binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(add(coeff(p, x, j), j=0..i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Aug 13 2015

CC[t_, n_] := Sum[2^(t - m)*StirlingS2[t, m], {m, 0, n}];
Table[CC[t, n], {t, 0, 12}, {n, 0, t}] // Flatten
(* Second program: *)
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[x^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!*Sum[Binomial[i, 2*k], {k, 0, i/2}]^j*b[n - i*j, i - 1], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, n} ] ][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

G.f.: sum(t>=0, n>=0, C_t(n)x^t/t!y^n) = exp(y/2 (exp(2*x)-1))/(1-y).

C_t(n) = Sum_{i=0..n} A075497(t,i).

A331610 Expansion of e.g.f.: exp(1 / (1 - tan(x)) - 1).

Original entry on oeis.org

1, 1, 3, 15, 97, 777, 7379, 80983, 1007137, 13986289, 214383171, 3593224767, 65347120705, 1281151315641, 26928292883795, 603928982033863, 14392387319349697, 363135896514611041, 9669298448057196291, 270932711729869233903, 7967970654277850949025

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..427

Cf. A000111, A000828, A003707, A004211, A006229, A331607.

S:= series(exp(1/(1-tan(x))-1), x, 31):
seq(coeff(S,x,i)*i!, i=0..30); # Robert Israel, Dec 10 2024

nmax = 20; CoefficientList[Series[Exp[1/(1 - Tan[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A000111[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1])/(n + 1)]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] 2^(k - 1) A000111[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

E.g.f.: exp(sin(x) / (cos(x) - sin(x))).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * 2^(k-1) * A000111(k) * a(n-k).
a(n) ~ 2^(2*n - 1/4) * exp(1/Pi - 1/2 + 2^(3/2)*sqrt(n/Pi) - n) * n^(n - 1/4) / Pi^(n + 1/4). - Vaclav Kotesovec, Jan 27 2020

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

A166922 E.g.f. exp(-x)exp(exp(2x)/2-1/2)/2 + 1/2.

Original entry on oeis.org

1, 0, 1, 2, 10, 48, 276, 1768, 12552, 97408, 818704, 7396384, 71380640, 732058880, 7943068992, 90833753728, 1091134058624, 13728139694080, 180436251140352, 2471790031618560

Offset: 0

Karol A. Penson, Oct 23 2009

nonn

In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - Vaclav Kotesovec, Jun 28 2022

G. C. Greubel, Table of n, a(n) for n = 0..250
R. Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

With[{nn = 25}, CoefficientList[Series[Exp[-t]*Exp[Exp[2*t]/2 - 1/2]/2 + 1/2, {t, 0, nn}], t] Range[0, nn]!] (* G. C. Greubel, May 28 2016 *)

x='x+O('x^66); Vec(serlaplace(exp(-x)*exp(exp(2*x)/2-1/2)/2+1/2)) \\ Joerg Arndt, May 06 2013

def A166922_list(n):  # n>=1
    T = [0]*(n+1); R = [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a + 2*(k*(b+c)+c)
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u; R.append(u/2)
    return R
A166922_list(20)
# Peter Luschny, Nov 01 2012

A004211(n) = -1 + 2*sum(k=0..n, C(n,k)*a(k)). - Peter Luschny, Nov 01 2012
G.f.: 1/2 + 1/2/Q(0), where Q(k)= 1 - 2*x*k - 2*x^2*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 06 2013
a(n) ~ 2^(n - 3/2) * n^(n - 1/2) * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022

Definition corrected on a suggestion of M. F. Hasler, Peter Luschny, Nov 05 2012

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).

A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 27, 83, 271, 971, 3865, 16879, 78985, 388385, 1987201, 10561385, 58443891, 337724057, 2040085491, 12862712499, 84357800063, 573182197539, 4021203303593, 29062345301487, 216129411635057, 1653180368063361, 13003920016983361, 105158133803473329

Offset: 0

Ilya Gutkovskiy, Feb 08 2022

nonn

Shifts 3 places left under 2nd-order binomial transform.

Cf. A000996, A000998, A004211, A007472, A210540, A351343, A351344, A351345.

nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 3, 1, Sum[Binomial[n - 3, k] 2^k a[n - k - 3], {k, 0, n - 3}]]; Table[a[n], {n, 0, 27}]

a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * 2^k * a(n-k-3).

A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 9, 27, 81, 245, 761, 2493, 8849, 34519, 147057, 670327, 3198561, 15732905, 79174929, 407127897, 2145061729, 11635963499, 65309080185, 380583443187, 2304629301041, 14475031232285, 93943897651017, 627220447621973, 4290783719133041, 29988917377046207

Offset: 0

Ilya Gutkovskiy, Feb 08 2022

nonn

Shifts 4 places left under 2nd-order binomial transform.

Cf. A004211, A007472, A010748, A210541, A275934, A351342, A351344, A351345.

nmax = 29; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 4, 1, Sum[Binomial[n - 4, k] 2^k a[n - k - 4], {k, 0, n - 4}]]; Table[a[n], {n, 0, 29}]

a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * 2^k * a(n-k-4).

A351344 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 9, 27, 81, 243, 731, 2223, 6939, 22727, 79971, 306929, 1282815, 5744361, 26984415, 130656409, 644739377, 3224303841, 16318576681, 83717193681, 436948772697, 2331807007139, 12791837178265, 72472130039123, 425239734375217, 2584950704996379

Offset: 0

Ilya Gutkovskiy, Feb 08 2022

nonn

Shifts 5 places left under 2nd-order binomial transform.

Cf. A004211, A007472, A010749, A210542, A275935, A351342, A351343, A351345.

nmax = 30; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 5, 1, Sum[Binomial[n - 5, k] 2^k a[n - k - 5], {k, 0, n - 5}]]; Table[a[n], {n, 0, 30}]

a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} binomial(n-5,k) * 2^k * a(n-k-5).

A351345 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 9, 27, 81, 243, 729, 2189, 6601, 20141, 63009, 205989, 718905, 2720543, 11183601, 49321367, 228895201, 1097860903, 5371546897, 26598018425, 132755261681, 667027581401, 3376011676481, 17249045903945, 89270689572497, 470069622480667

Offset: 0

Ilya Gutkovskiy, Feb 08 2022

nonn

Shifts 6 places left under 2nd-order binomial transform.

Cf. A004211, A007472, A010750, A210543, A275936, A351342, A351343, A351344

nmax = 31; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 6, 1, Sum[Binomial[n - 6, k] 2^k a[n - k - 6], {k, 0, n - 6}]]; Table[a[n], {n, 0, 31}]

a(0) = ... = a(5) = 1; a(n) = Sum_{k=0..n-6} binomial(n-6,k) * 2^k * a(n-k-6).

cp's OEIS Frontend

A241579 Square array read by antidiagonals downwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).

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A261275 Number of set partitions C_t(n) of {1,2,...,t} into at most n parts, with an even number of elements in each part distinguished by marks; triangle C_t(n), t>=0, 0<=n<=t, read by rows.

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A331610 Expansion of e.g.f.: exp(1 / (1 - tan(x)) - 1).

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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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A166922 E.g.f. exp(-x)*exp(exp(2*x)/2-1/2)/2 + 1/2.

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PARI

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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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A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2*x)) / (1 - 2*x).

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A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2*x)) / (1 - 2*x).

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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.

A166922 E.g.f. exp(-x)exp(exp(2x)/2-1/2)/2 + 1/2.

A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2x)) / (1 - 2x).

A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2x)) / (1 - 2x).

A351344 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x/(1 - 2x)) / (1 - 2x).

A351345 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 * A(x/(1 - 2x)) / (1 - 2x).