A002866 -id:A002866 - cp's OEIS frontend

A051583 a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).

1, 11, 143, 2145, 36465, 692835, 14549535, 334639305, 8365982625, 225881530875, 6550564395375, 203067496256625, 6701227376468625, 234542958176401875, 8678089452526869375, 338445488648547905625, 13876265034590464130625

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..398
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A178647.

Cf. A051577, A051578, A051579, A051580, A051581, A051582 (rows m=0..8).

List([0..20], n-> Product([0..n-1], j-> 2*j+11) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+11: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

seq(2^n*pochhammer(11/2,n), n = 0..20); # G. C. Greubel, Nov 12 2019

(2*Range[0,20]+9)!!/945 (* Harvey P. Dale, Apr 10 2019 *)
Table[2^n*Pochhammer[11/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=0,n-2, 2*j+11) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+11) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+9)!!/9!!.
E.g.f.: 1/(1-2*x)^(11/2).
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 11)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 11*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 11*x/(1 - 13*x/(1 - 2*x/(1 - 15*x/(1 - 4*x/(1 - 17*x/(1 - 6*x/(1 - ... - (2*n + 11)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 945 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1332, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 945 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 684, where erfi is the imaginary error function. (End)

A035101 E.g.f. x(c(x/2)-1)/(1-2x), where c(x) = g.f. for Catalan numbers A000108.

Original entry on oeis.org

0, 1, 9, 87, 975, 12645, 187425, 3133935, 58437855, 1203216525, 27125492625, 664761133575, 17600023616175, 500706514833525, 15234653491682625, 493699195087473375, 16977671416936605375, 617528830880480644125, 23687738668934964248625

Offset: 1

Wolfdieter Lang

2nd column of triangular array A035342 whose first column is given by A001147(n), n >= 1. Recursion: a(n) = 2*n*a(n-1)+ A001147(n-1), n >= 2, a(1)=0.
a(n) gives the number of organically labeled forests (sets) with two rooted ordered trees with n non-root vertices. See the example a(3)=9 given in A035342. Organic labeling means that the vertex labels along the (unique) path from the root to any of the leaves (degree 1, non-root vertices) is increasing. - Wolfdieter Lang, Aug 07 2007
a(n), n>=2, enumerates unordered n-vertex forests composed of two plane (ordered) ternary (3-ary) trees with increasing vertex labeling. See A001147 (number of increasing ternary trees) and a D. Callan comment there. For a picture of some ternary trees see a W. Lang link under A001764.
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly 1 of the remaining n-1 chords are contained within the marked chord, see [Young]. - Donovan Young, Aug 11 2020

			a(2)=1 for the forest: {r1-1, r2-2} (with root labels r1 and r2). The order between the components of the forest is irrelevant (like for sets).
a(3)=9 increasing ternary 2-forest with n=3 vertices: there are three 2-forests (the one vertex tree together with any of the three different 2-vertex trees) each with three increasing labelings. - _Wolfdieter Lang_, Sep 14 2007

Robert Israel, Table of n, a(n) for n = 1..370
Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Cf. A000108, A002866, A008549, A336599.

Cf. A001147 (m=1 column of A035342). See a D. Callan comment there on the number of increasing ordered rooted trees on n+1 vertices.

I:=[0,1,9]; [n le 3 select I[n] else - 2*(n-1)*(2*n-3)*Self(n-2)+(4*n-3)*Self(n-1): n in [1..30]]; // Vincenzo Librandi, Sep 12 2015

F:= gfun:-rectoproc({(4*n^2+6*n+2)*a(n)+(-4*n-5)*a(n+1)+a(n+2),a(1)=0,a(2)=1,a(3)=9},a(n),remember):
map(f, [$1..30]); # Robert Israel, Sep 11 2015

Table[Round [n! (4^(n - 1) - Binomial[2 n, n]/2)/2^(n - 1)], {n, 1, 20}] (* Vincenzo Librandi, Sep 12 2015 *)

a(n) = n!*(4^(n-1)-binomial(2*n, n)/2)/2^(n-1);
vector(40, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n) = n!*A008549(n-1)/2^(n-1) = n!(4^(n-1)-binomial(2*n, n)/2)/2^(n-1).
a(n) = (2n-2)*a(n-1) + A129890(n-2). - Philippe Deléham, Oct 28 2013
a(n) = n!*2^(n-1) - A001147(n) = A002866(n) - A001147(n). - Peter Bala, Sep 11 2015
a(n) = -2*(n-1)*(2*n-3)*a(n-2)+(4*n-3)*a(n-1). - Robert Israel, Sep 11 2015

A081405 a(n) = (n+1)*a(n-2) with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 3, 4, 15, 24, 105, 192, 945, 1920, 10395, 23040, 135135, 322560, 2027025, 5160960, 34459425, 92897280, 654729075, 1857945600, 13749310575, 40874803200, 316234143225, 980995276800, 7905853580625, 25505877196800, 213458046676875

Offset: 0

Labos Elemer, Apr 01 2003

nonn

A001147 and A002866 combined.

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 15*x^4 + 24*x^5 + 105*x^6 + 192*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..790

Cf. A000142, A001147, A002866.

a:= function(n)
    if n<2 then return 1;
    else return (n+1)*a(n-2);
    fi;
  end;
List([0..30], n-> a(n) ); # G. C. Greubel, Aug 24 2019

[n le 1 select 1 else (n+1)*Self(n-1): n in [0..30]]; // Vincenzo Librandi, Oct 26 2014

a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-2]*(n+1)^2) od: seq(sqrt(a[n]), n=0..26); # Zerinvary Lajos, Mar 04 2008

f[n_]:= (n+1)*f[n-2]; f[0] = 1; f[1] = 1; Table[f[n], {n, 1, 30}]
a[ n_]:= If[ n < 0, 0, If[OddQ[n], 2^((n-1)/2) ((n+1)/2)!, (n+1)!!]]; (* Michael Somos, Jan 24 2014 *)
RecurrenceTable[{a[0]==a[1]==1,a[n]==(n+1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Nov 05 2021 *)

{a(n) = if( n<2, n>=0, (n+1) * a(n-2))}; /* Michael Somos, Jan 24 2014 */

{a(n) = if( n<0, 0, if( n%2, 2^(n\2) * (n\2 + 1)!, (n+1)! / (2^(n\2) * (n\2)!)))}; /* Michael Somos, Jan 24 2014 */

def a(n):
    if n<2: return 1
    else: return (n+1)*a(n-2)
[a(n) for n in (0..30)] # G. C. Greubel, Aug 24 2019

a(0)=a(1)=1; a(2n) = A001147(2*n-2) odd terms, double factorial numbers; a(2n-1) = A002866(n) = 2^(n-1)*n!
0 = a(n)*(a(n+1) - a(n+3)) + a(n+1)*a(n+2) if n>=0. - Michael Somos, Jan 24 2014
a(n) = (n-1)-st term of column 1 of the array at A249159, for n >= 0. - Clark Kimberling, Oct 23 2014

A144829 Partial products of successive terms of A017209; a(0)=1 .

Original entry on oeis.org

1, 4, 52, 1144, 35464, 1418560, 69509440, 4031547520, 270113683840, 20528639971840, 1744934397606400, 164023833375001600, 16894454837625164800, 1892178941814018457600, 228953651959496233369600, 29763974754734510338048000, 4137192490908096936988672000

Offset: 0

Philippe Deléham, Sep 21 2008

			a(0)=1, a(1)=4, a(2)=4*13=52, a(3)=4*13*22=1144, a(4)=4*13*22*31=35464, ...

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A001715, A002866, A007559, A008546, A047053.

Cf. A048994, A049308, A132393, A144827, A144828.

[n le 2 select 4^(n-1) else (9*n-14)*Self(n-1): n in [1..30]]; // G. C. Greubel, May 26 2022

Table[4*9^(n-1)*Pochhammer[13/9, n-1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 29 2021 *)

a(n) = (-5)^n*sum(k=0, n, (9/5)^k*stirling(n+1,n+1-k, 1)); \\ Michel Marcus, Feb 20 2015

[9^n*rising_factorial(4/9, n) for n in (0..30)] # G. C. Greubel, May 26 2022

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*9^(n-k).
a(n) = (-5)^n*Sum_{k=0..n} (9/5)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) + (5-9*n)*a(n-1) = 0. - R. J. Mathar, Sep 04 2016
From Vaclav Kotesovec, Nov 29 2021: (Start)
a(n) = 9^n * Gamma(n + 4/9) / Gamma(4/9).
a(n) ~ sqrt(2*Pi) * 9^n * n^(n - 1/18) / (Gamma(4/9) * exp(n)). (End)
From G. C. Greubel, May 26 2022: (Start)
G.f.: hypergeometric2F0([1, 4/9], [], 9*x).
E.g.f.: (1-9*x)^(-4/9). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/9^5)^(1/9)*(Gamma(4/9) - Gamma(4/9, 1/9)). - Amiram Eldar, Dec 21 2022

a(9) originally given incorrectly as 20520639971840 corrected by Peter Bala, Feb 20 2015

A156992 Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.

Original entry on oeis.org

1, 2, 2, 6, 12, 6, 24, 72, 72, 24, 120, 480, 720, 480, 120, 720, 3600, 7200, 7200, 3600, 720, 5040, 30240, 75600, 100800, 75600, 30240, 5040, 40320, 282240, 846720, 1411200, 1411200, 846720, 282240, 40320, 362880, 2903040, 10160640, 20321280, 25401600, 20321280, 10160640, 2903040, 362880

Offset: 1

Roger L. Bagula, Feb 20 2009

Partition {1,2,...,n} into m subsets, arrange (linearly order) the elements within each subset, then arrange the subsets. - Geoffrey Critzer, Mar 05 2010
From Dennis P. Walsh, Nov 26 2011: (Start)
Number of ways to arrange n different books in a k-shelf bookcase leaving no shelf empty.
There are n! ways to arrange the books in one long line. With ni denoting the number of books for shelf i, we have n = n1 + n2 + ... + nk. Since the number of compositions of n with k summands is binomial(n-1,k-1), we obtain T(n,k) = n!*binomial(n-1,k-1) for the number of ways to arrange the n books on the k shelves.
Equivalently, T(n,k) is the number of ways to stack n different alphabet blocks into k labeled stacks.
Also, T(n,k) is the number of injective functions f:[n]->[n+k] such that (i) the pre-image of (n+j) exists for j=1..k and (ii) f has no fixed points, that is, for all x, f(x) does not equal x.
T(n,k) is the number of labeled, rooted forests that have (i) exactly k roots, (ii) each root labeled larger than any nonroot, (iii) each root with exactly one child node, (iv) n non-root nodes, and (v) at most one child node for each node in the forest.
(End)
Essentially, the triangle given by (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 29 2011
T(n,j+k) = Sum_{i=j..n-k} binomial(n,i)*T(i,j)*T(n-i,k). - Dennis P. Walsh, Nov 29 2011

			The triangle starts:
      1;
      2,      2;
      6,     12,      6;
     24,     72,     72,      24;
    120,    480,    720,     480,     120;
    720,   3600,   7200,    7200,    3600,    720;
   5040,  30240,  75600,  100800,   75600,  30240,   5040;
  40320, 282240, 846720, 1411200, 1411200, 846720, 282240, 40320;
From _Dennis P. Walsh_, Nov 26 2011: (Start)
T(3,2) = 12 since there are 12 ways to arrange books b1, b2, and b3 on shelves <shelf1><shelf2>:
   <b1><b2,b3>, <b1><b3,b2>, <b2><b1,b3>, <b2><b3,b1>,
   <b3><b1,b2>, <b3><b2,b1>, <b2,b3><b1>, <b3,b2><b1>,
   <b1,b3><b2>, <b3,b1><b2>, <b1,b2><b3>, <b2,b1><b3>.
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 98

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers

Cf. A002866 (row sums).
Column 1 = A000142. Column 2 = A001286 * 2! = A062119. Column 3 = A001754 * 3!. Column 4 = A001755 * 4!. Column 5 = A001777 * 5!. Column 6 = A001778 * 6!. Column 7 = A111597 * 7!. Column 8 = A111598 * 8!. Cf. A105278. - Geoffrey Critzer, Mar 05 2010
T(2n,n) gives A123072.

[Factorial(n)*Binomial(n-1,k-1): k in [1..n], n in [1..10]]; // G. C. Greubel, May 10 2021

seq(seq(n!*binomial(n-1,k-1),k=1..n),n=1..10); # Dennis P. Walsh, Nov 26 2011
with(PolynomialTools): p := (n,x) -> (n+1)!*hypergeom([-n],[],-x);
seq(CoefficientList(simplify(p(n,x)),x),n=0..5); # Peter Luschny, Apr 08 2015

Table[n!*Binomial[n-1, k-1], {n,10}, {k,n}]//Flatten

flatten([[factorial(n)*binomial(n-1,k-1) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, May 10 2021

E.g.f. for column k is (x/(1-x))^k. - Geoffrey Critzer, Mar 05 2010
T(n,k) = A000142(n)*A007318(n-1,k-1). - Dennis P. Walsh, Nov 26 2011
Coefficient triangle of the polynomials p(n,x) = (n+1)!*hypergeom([-n],[],-x). - Peter Luschny, Apr 08 2015

A370419 A(n, k) = 2^n*Pochhammer(k/2, n). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 8, 3, 1, 0, 105, 48, 15, 4, 1, 0, 945, 384, 105, 24, 5, 1, 0, 10395, 3840, 945, 192, 35, 6, 1, 0, 135135, 46080, 10395, 1920, 315, 48, 7, 1, 0, 2027025, 645120, 135135, 23040, 3465, 480, 63, 8, 1

Offset: 0

Peter Luschny, Mar 04 2024

			The array starts:
[0] 1,   1,    1,     1,     1,     1,     1,      1,      1, ...
[1] 0,   1,    2,     3,     4,     5,     6,      7,      8, ...
[2] 0,   3,    8,    15,    24,    35,    48,     63,     80, ...
[3] 0,  15,   48,   105,   192,   315,   480,    693,    960, ...
[4] 0, 105,  384,   945,  1920,  3465,  5760,   9009,  13440, ...
[5] 0, 945, 3840, 10395, 23040, 45045, 80640, 135135, 215040, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,   1;
[2] 0,   1,   1;
[3] 0,   3,   2,   1;
[4] 0,  15,   8,   3,  1;
[5] 0, 105,  48,  15,  4, 1;
[6] 0, 945, 384, 105, 24, 5, 1;
.
From _Werner Schulte_, Mar 07 2024: (Start)
Illustrating the LU decomposition of A:
    / 1                \   / 1 1 1 1 1 ... \   / 1   1   1   1    1 ... \
    | 0   1            |   |   1 2 3 4 ... |   | 0   1   2   3    4 ... |
    | 0   3   2        | * |     1 3 6 ... | = | 0   3   8  15   24 ... |
    | 0  15  18   6    |   |       1 4 ... |   | 0  15  48 105  192 ... |
    | 0 105 174 108 24 |   |         1 ... |   | 0 105 384 945 1920 ... |
    | . . .            |   | . . .         |   | . . .                  |. (End)

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Columns: A000007, A001147, A000165, A001147 (shifted), A002866, A051577, A051578, A051579, A051580.
Rows: A000012, A001477, A005563, A370912, A190577.
Cf. A035342, A265649, A370890, A370982 (row sums of the triangle), A370915, A371025, A371077.

A := (n, k) -> 2^n*pochhammer(k/2, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 2*x)^(-k/2);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..8);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-2)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the comment of Werner Schulte about the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371025(n - 1,  k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U));  #  Peter Luschny, Mar 08 2024

A370419[n_, k_] := 2^n*Pochhammer[k/2, n];
Table[A370419[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 06 2024 *)

def A(n, k): return 2**n * rising_factorial(k/2, n)
for n in range(6): print([A(n, k) for k in range(9)])

The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-2)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
From Werner Schulte, Mar 07 2024: (Start)
A(n, k) = Product_{i=1..n} (2*i - 2 + k).
E.g.f. of column k: Sum_{n>=0} A(n, k) * t^n / (n!) = (1/sqrt(1 - 2*t))^k.
A(n, k) = A(n+1, k-2) / (k - 2) for k > 2.
A(n, k) = Sum_{i=0..k-1} i! * A265649(n, i) * binomial(k-1, i) for k > 0.
E.g.f. of row n > 0: Sum_{k>=1} A(n, k) * x^k / (k!) = (Sum_{k=1..n} A035342(n, k) * x^k) * exp(x).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (k! * n!) = exp(x/sqrt(1 - 2*t)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1 / (1 - x/sqrt(1 - 2*t)).
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L, where L is defined L(n, k) = A035342(n, k) * k! for 1 <= k <= n and L(n, 0) = 0^n. Note that L(n, k) + L(n, k+1) = A265649(n, k) * k! for 0 <= k <= n. (End)

A131222 Exponential Riordan array [1, log((1-x)/(1-2x))].

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 14, 9, 1, 0, 90, 83, 18, 1, 0, 744, 870, 275, 30, 1, 0, 7560, 10474, 4275, 685, 45, 1, 0, 91440, 143892, 70924, 14805, 1435, 63, 1, 0, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1

Offset: 0

Paul Barry, Jun 18 2007

This is also the matrix product of the unsigned Lah numbers and the Stirling cycle numbers. See also A079639 and A079640 for variants based on an (1,1)-offset of the number triangles. - Peter Luschny, Apr 12 2015
The Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016
Essentially the same as A079638. - Peter Bala, Feb 12 2022

			Number triangle starts:
  1,
  0,   1;
  0,   3,   1;
  0,  14,   9,   1;
  0,  90,  83,  18,  1;
  0, 744, 870, 275, 30,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Paul Barry, Exponential Riordan arrays and permutation enumeration,Journal of Integer Sequences, Vol. 13 (2010).

Cf. A000007, A002866, A029767, A079638, A079639, A079640.

RioExp := (d,h,n,k) -> coeftayl(d*h^k, x=0,n)*n!/k!:
A131222 := (n,k) -> RioExp(1,log((1-x)/(1-2*x)),n,k):
seq(print(seq(A131222(n,k),k=0..n)),n=0..5); # Peter Luschny, Apr 15 2015
# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0,1,n!*(2^(n+1)-1)), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 1, #! (2^(#+1) - 1)]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

T(n,m):=if n=0 and m=0 then 1 else n!*sum((stirling1(k,m)*2^(n-k)*binomial(n-1,k-1))/k!,k,m,n); /* Vladimir Kruchinin, Sep 27 2012 */

def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number1) # as a square matrix Peter Luschny, Apr 12 2015
# alternatively:

# uses[bell_matrix from A264428]
bell_matrix(lambda n: A029767(n+1), 10) # Peter Luschny, Jan 18 2016

Row sums are A002866.
Second column is A029767.
T(n,m) = n! * Sum_{k=m..n} Stirling1(k,m)*2^(n-k)*binomial(n-1,k-1)/k!, n >= m >= 0. - Vladimir Kruchinin, Sep 27 2012

A308876 Expansion of e.g.f. exp(x)(1 - x)/(1 - 2x).

Original entry on oeis.org

1, 2, 7, 40, 317, 3166, 37987, 531812, 8508985, 153161722, 3063234431, 67391157472, 1617387779317, 42052082262230, 1177458303342427, 35323749100272796, 1130359971208729457, 38432239021096801522, 1383560604759484854775, 52575302980860424481432

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Binomial transform of A002866.

Alois P. Heinz, Table of n, a(n) for n = 0..403

Cf. A002866, A010844, A011782, A067273, A161936, A271705, A294466, A327606.

Row sums of A326659.

a:= n-> n! * add(ceil(2^(n-k-1))/k!, k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 2019

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

a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 2^(k-1) * k!.
a(n) = A010844(n) - A067273(n).
a(n) ~ n! * 2^(n-1) * exp(1/2). - Vaclav Kotesovec, Jun 29 2019
a(n) = Sum_{k=0..n} k! * A271705(n,k). - Alois P. Heinz, Sep 12 2019

A364324 a(n) = n!*tribonacci(n+2).

Original entry on oeis.org

1, 1, 4, 24, 168, 1560, 17280, 221760, 3265920, 54069120, 994291200, 20118067200, 444034483200, 10617070464000, 273391121203200, 7542665754624000, 221969877921792000, 6940528784437248000, 229781192298577920000, 8030036368187817984000, 295390797322766745600000

Offset: 0

Enrique Navarrete, Jul 18 2023

nonn

a(n) is the number of ways to partition [n] into blocks of size at most 3, order the blocks, and order the elements within each block.

			a(5) = 1560 since the number of ways to partition [5] into blocks of size at most 3, order the blocks, and order the elements within each block are the following:
1) 1,2,3,4,5: 120 ordered blocks; 120 ways;
2) 12,3,4,5: 240 ordered blocks; 480 ways;
3) 12,34,5: 90 ordered blocks; 360 ways;
4) 123,45: 20 ordered blocks; 240 ways;
5) 123,4,5: 60 ordered blocks; 360 ways.

Cf. A000073, A000142, A002866, A005442, A189886.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-i)*binomial(n, i)*i!, i=1..min(n, 3)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 18 2023

With[{m = 21}, Range[0, m - 1]! * LinearRecurrence[{1, 1, 1}, {1, 1, 2}, m]] (* Amiram Eldar, Jul 28 2023 *)

E.g.f.: 1/(1-x-x^2-x^3).

a(n) = A000142(n) * A000073(n+2).

A167594 A triangle related to the GF(z) formulas of the rows of the ED4 array A167584.

Original entry on oeis.org

1, 2, 2, 9, 2, 13, 60, -12, 68, 76, 525, -300, 774, 132, 789, 5670, -5250, 11820, -3636, 6702, 7734, 72765, -92610, 212415, -143340, 143307, 19086, 110937, 1081080, -1746360, 4286520, -4246200, 4156200, -1204200, 1305000, 1528920

Offset: 1

Johannes W. Meijer, Nov 10 2009

The GF(z) formulas given below correspond to the first ten rows of the ED4 array A167584. The polynomials in their numerators lead to the triangle given above.

			Row 1: GF(z) = 1/(1-z).
Row 2: GF(z) = (2*z + 2)/(1-z)^2.
Row 3: GF(z) = (9*z^2 + 2*z + 13)/(1-z)^3.
Row 4: GF(z) = (60*z^3 - 12*z^2 + 68*z + 76)/(1-z)^4.
Row 5: GF(z) = (525*z^4 - 300*z^3 + 774*z^2 + 132*z + 789)/(1-z)^5.
Row 6: GF(z) = (5670*z^5 - 5250*z^4 + 11820*z^3 - 3636*z^2 + 6702*z + 7734)/(1-z)^6.
Row 7: GF(z) = (72765*z^6 - 92610*z^5 + 212415*z^4 - 143340*z^3 + 143307*z^2 + 19086*z + 110937)/ (1-z)^7.
Row 8: GF(z) = (1081080*z^7 - 1746360*z^6 + 4286520*z^5 - 4246200*z^4 + 4156200*z^3 - 1204200*z^2 + 1305000*z + 1528920)/(1-z)^8.
Row 9: GF(z) = (18243225*z^8 - 35675640*z^7 + 95176620*z^6 -121723560*z^5 + 132769350*z^4 - 73816200*z^3 + 45017100*z^2 + 4887720*z + 28018665) / (1-z)^9.
Row 10: GF(z) = (344594250*z^9 - 790539750*z^8 + 2299457160*z^7 - 3567314520*z^6 + 4441299660*z^5 - 3398138100*z^4 + 2160066600*z^3 - 550619640*z^2 + 421244730*z + 497895210)/(1-z)^10.

A167584 is the ED4 array.
A001193 equals the first left hand column.
A024199 equals the first right hand column.
A002866 equals the row sums.

cp's OEIS Frontend

A051583 a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A035101 E.g.f. x*(c(x/2)-1)/(1-2*x), where c(x) = g.f. for Catalan numbers A000108.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A081405 a(n) = (n+1)*a(n-2) with a(0) = a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A144829 Partial products of successive terms of A017209; a(0)=1 .

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A156992 Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A370419 A(n, k) = 2^n*Pochhammer(k/2, n). Square array read by ascending antidiagonals.

Views

Author

Keywords

A035101 E.g.f. x(c(x/2)-1)/(1-2x), where c(x) = g.f. for Catalan numbers A000108.

A308876 Expansion of e.g.f. exp(x)(1 - x)/(1 - 2x).