A052852 -id:A052852 - cp's OEIS frontend

A102290 Total number of even lists in all sets of lists, cf. A000262.

0, 0, 2, 6, 60, 380, 3990, 37002, 450296, 5373720, 76018410, 1096730030, 17814654132, 299645294676, 5511836578430, 105550556136690, 2171244984679920, 46545825736022192, 1059273836225051346, 25100215228045842390, 626204775725372971820, 16239127347086448236460

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A052852, A059570, A066897, A066898, A081358, A092691.

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[0,0]cat[Factorial(n)*(&+[(-1)^(n+j)*l(j,-1): j in [0..n-2]]): n in [2..30]]; // G. C. Greubel, Mar 09 2021

Gser:=series(x^2*exp(x/(1-x))/(1-x^2),x=0,22):seq(n!*coeff(Gser,x^n),n=1..21); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::even, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

Rest[CoefficientList[Series[x^2/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
Table[If[n<2, 0, n!*Sum[(-1)^(n-j)*LaguerreL[j, -1], {j,0,n-2}]], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[0,0]+[factorial(n)*sum((-1)^(n+j)*gen_laguerre(j,0,-1) for j in (0..n-2)) for n in (2..30)] # G. C. Greubel, Mar 09 2021

E.g.f.: x^2/(1-x^2)*exp(x/(1-x)).
Recurrence: (n-2)*a(n) = (n-2)*n*a(n-1) + (n-1)^2*n*a(n-2) - (n-3)*(n-2)*(n-1)*n*a(n-3). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 - 41/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
a(n) = n! * Sum_{j=0..n-2} (-1)^(n+j)*LaguerreL(j, -1) for n>1 with a(0)=a(1)=0. - G. C. Greubel, Mar 09 2021

More terms from Emeric Deutsch, Mar 27 2005

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A152151 Riordan array [(1-x)exp(x/(1-x)),x].

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 21, 16, 6, 0, 1, 136, 105, 40, 10, 0, 1, 1045, 816, 315, 80, 15, 0, 1, 9276, 7315, 2856, 735, 140, 21, 0, 1, 93289, 74208, 29260, 7616, 1470, 224, 28, 0, 1, 1047376, 839601, 333936, 87780, 17136, 2646, 336, 36

Offset: 0

Paul Barry, Nov 26 2008

First column is essentially A052852.

			Triangle begins
1,
0, 1,
1, 0, 1,
4, 3, 0, 1,
21, 16, 6, 0, 1,
136, 105, 40, 10, 0, 1,
1045, 816, 315, 80, 15, 0, 1,
9276, 7315, 2856, 735, 140, 21, 0, 1,
with production matrix which begins
0, 1,
1, 0, 1,
4, 2, 0, 1,
18, 12, 3, 0, 1,
96, 72, 24, 4, 0, 1,
600, 480, 180, 40, 5, 0, 1;
4320, 3600, 1440, 360, 60, 6, 1
with e.g.f. exp(x*t)(t+1/(1-x)^2).

A173227 Partial sums of A000262.

Original entry on oeis.org

1, 2, 5, 18, 91, 592, 4643, 42276, 436629, 5033182, 63974273, 888047414, 13358209647, 216334610860, 3751352135263, 69325155322184, 1359759373992105, 28206375825238458, 616839844140642301, 14181213537729200474, 341879141423814854915, 8623032181189674581256

Offset: 0

Jonathan Vos Post, Feb 13 2010

nonn

Partial sums of the number of "sets of lists": number of partitions of {1,..,n} into any number of lists, where a list means an ordered subset. The subsequence of primes begins: 2, 5, 4643, 616839844140642301.

			a(20) = 1 + 1 + 3 + 13 + 73 + 501 + 4051 + 37633 + 394353 + 4596553 + 58941091 + 824073141 + 12470162233 + 202976401213 + 3535017524403 + 65573803186921 + 1290434218669921 + 26846616451246353 + 588633468315403843 + 13564373693588558173 + 327697927886085654441.

Alois P. Heinz, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A001263, A001700, A002868, A008297, A052852, A066668, A111596,.

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[n eq 0 select 1 else 1 + (&+[ Factorial(j)*( l(j,-1) - l(j-1,-1) ): j in [1..n]]): n in [0..25]]; // G. C. Greubel, Mar 09 2021

b:= proc(n) option remember; `if`(n=0, 1, add(
       b(n-j)*j!*binomial(n-1, j-1), j=1..n))
    end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

With[{m = 25}, CoefficientList[Exp[x/(1-x)] + O[x]^m, x] Range[0, m-1]!// Accumulate] (* Jean-François Alcover, Nov 21 2020 *)
Table[1 +Sum[j!*(LaguerreL[j, -1] -LaguerreL[j-1, -1]), {j,n}], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[1 + sum(factorial(j)*(gen_laguerre(j,0,-1) - gen_laguerre(j-1,0,-1)) for j in (1..n)) for n in (0..30)] # G. C. Greubel, Mar 09 2021

From Vaclav Kotesovec, Oct 25 2016: (Start)
a(n) = 2*n*a(n-1) - (n^2 - n + 1)*a(n-2) + (n-2)*(n-1)*a(n-3).
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/sqrt(2) * (1 - 5/(48*sqrt(n))).
(End)
a(n) = 1 + Sum_{j=1..n} j!*( LaguerreL(j,-1) - LaguerreL(j-1,-1) ). - G. C. Greubel, Mar 09 2021

A300159 Number of ways of converting one set of lists containing n elements to another set of lists containing n elements by removing the last element from one of the lists and either appending it to an existing list or treating it as a new list.

Original entry on oeis.org

0, 0, 4, 30, 240, 2140, 21300, 235074, 2853760, 37819800, 543445380, 8416452550, 139753069104, 2476581106740, 46648575724660, 930581784937770, 19597766647728000, 434455097953799344, 10112163333554834820, 246539064280189932270, 6282671083849941925360

Offset: 0

Mitchell Keith Bloch, Feb 26 2018

nonn

All terms are even.

			a(0) = 0 since for 0 lists, 0 conversions are possible.
a(1) = 0 since for the 1 set of 1 list of length 1, there exist no possible conversions.
a(2) = 4 since for the 2 sets of 1 list of length 2, there exists only 1 conversion, and for the 1 set of 2 lists of length 1, there exist 2 conversions.
a(3) = 30 since for the 6 sets of 1 list of length 3, there exists 1 conversion, for the 6 sets of 1 list of length 2 and 1 list of length 1, there exist 3 conversions, and for the 1 set of 3 lists of length 1, there exists 6 conversions.

Alois P. Heinz, Table of n, a(n) for n = 0..443 (first 71 terms from Mitchell Keith Bloch)
Mitchell Keith Bloch, Program C++
Mitchell Keith Bloch, Program C++ with Boost
Index entries for sequences related to Laguerre polynomials

Extends A000262 to count conversions in addition to sets of lists.

Cf. A000041, A006152, A052852, A103194.

l:= func< n,b | Evaluate(LaguerrePolynomial(n,1), b) >;
[0,0,4]cat[Factorial(n)*( 2*l(n-2,-1) - l(n-3,-1) ): n in [3..30]]; // G. C. Greubel, Mar 09 2021

b:= proc(n, t, c) option remember; `if`(n=0, t^2-c, add(j!*
      binomial(n-1, j-1)*b(n-j, t+1, c+`if`(j=1, 1, 0)), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 05 2018
# second Maple program:
a:= proc(n) option remember; `if`(n<6, [0$2, 4, 30, 240, 2140][n+1],
     (n*(2*n^2-13*n+16)*a(n-1)-n*(n-1)*(n-3)*(n-4)*a(n-2))/((n-2)*(n-5)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 05 2018

(* First program *)
b[n_, t_, c_]:= b[n,t,c]= If[n==0, t^2 -c, Sum[j! Binomial[n-1, j-1]b[n-j,t+1,c + If[j==1, 1, 0]], {j,n}]];
a[n_]:= b[n, 0, 0];
a/@ Range[0, 25] (* Jean-François Alcover, Nov 24 2020, after Alois P. Heinz *)
(* Second program *)
Table[If[n<2, 0, n!*(2*LaguerreL[n-2,1,-1] -LaguerreL[n-3,1,-1])], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[0,0,4]+[factorial(n)*(2*gen_laguerre(n-2,1,-1) - gen_laguerre(n-3,0,-1)) for n in (3..30)] # G. C. Greubel, Mar 09 2021

a(n) = Sum_{i=1..p(n)} (n!/(Product_{j=1..n} k(i,j)!) * ((Sum_{j=1..n} k(i,j))^2 - k(i,1))) (where p(n) is the number of partitions A000041 and k(i,j) is the number of partitions of size j in partitioning i).
From Alois P. Heinz, Mar 05 2018: (Start)
E.g.f.: x^2*(2-x)*exp(x/(1-x))/(x-1)^2.
a(n) = (n*(2*n^2-13*n+16)*a(n-1) - n*(n-1)*(n-3)*(n-4)*a(n-2)) / ((n-2)*(n-5)) for n>5. (End)
a(n) ~ n^(n + 3/4) * exp(2*sqrt(n) - n - 1/2) / sqrt(2). - Vaclav Kotesovec, Jun 02 2018
a(n) = n!*( 2*LaguerreL(n-2,1,-1) - LaguerreL(n-3,1,-1) ) for n > 1, with a(0) = a(1) = 0. - G. C. Greubel, Mar 09 2021

More terms from Mitchell Keith Bloch, Mar 05 2018

A316666 Number of simple relaxed compacted binary trees of right height at most one with no sequences on level 1 and no final sequences on level 0.

Original entry on oeis.org

1, 0, 1, 3, 15, 87, 597, 4701, 41787, 413691, 4512993, 53779833, 695000919, 9680369943, 144560191149, 2303928046437, 39031251610227, 700394126116851, 13270625547477177, 264748979672169681, 5547121478845459983, 121784530649198053263, 2795749225338111831429, 66981491857058929294653

Offset: 0

Michael Wallner, Jul 10 2018

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and at most n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. It is called simple if for nodes with two pointers both point to the same node. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. See the Wallner link.

a(n) is one of two "basis" sequences for sequences of the form a(0)=a, a(1)=b, a(n) = n*a(n-1) + (n-1)*a(n-2), the second basis sequence being A096654 (with 0 appended as a(0)). The sum of these sequences is listed as A000255. - Gary Detlefs, Dec 11 2018

G. C. Greubel, Table of n, a(n) for n = 0..448
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952, A288953 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of simple relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000255, A096654.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3*Exp(-x) + x-2)/(1-x)^2 )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 12 2018

aseq := n-> 3*round((n+2)*n!/exp(1))-(n+2)*n!: bseq := n-> (n+2)*n!- 2* round((n+2)*n!/exp(1)): s := (a,b,n)-> a*aseq(n) + b*bseq( n): seq(s(1,0,n),n = 0..20);  # Gary Detlefs, Dec 11 2018

terms = 24;
CoefficientList[(3E^-z+z-2)/(1-z)^2 + O[z]^terms, z] Range[0, terms-1]! (* Jean-François Alcover, Sep 14 2018 *)

Vec(serlaplace((3*exp(-x + O(x^25)) + x - 2)/(1 - x)^2)) \\ Andrew Howroyd, Jul 10 2018

E.g.f.: (3*exp(-z)+z-2)/(1-z)^2.
a(n) ~ (3*exp(-1) - 1) * n * n!. - Vaclav Kotesovec, Jul 12 2018
a(n) = 3*round((n+2)*n!/e) - (n+2)*n!. - Gary Detlefs, Dec 11 2018
From Seiichi Manyama, Apr 25 2025: (Start)
a(n) = 3 * A000255(n) - n! - (n+1)!.
a(0) = 1, a(1) = 0; a(n) = n*a(n-1) + (n-1)*a(n-2). (End)

A317096 Expansion of e.g.f. ((1 - x)/(1 - 2x))exp(x/(x - 1)).

Original entry on oeis.org

1, 0, 1, 8, 69, 704, 8485, 118824, 1900297, 34191296, 683657001, 15038537480, 360903291661, 9383240195328, 262727926084429, 7881806223689384, 252217461390469905, 8575390623429206144, 308714050531090308817, 11731134397549023854856, 469245396934886909801941, 19708307298664103361642560

Offset: 0

Ilya Gutkovskiy, Aug 01 2018

nonn

Lah transform of A000166.

G. C. Greubel, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Transforms
Index entries for sequences related to Laguerre polynomials

Cf. A000166, A002866, A052852, A059115.

R:=PowerSeriesRing(Rationals(), 26);
Coefficients(R!(Laplace( ((1-x)/(1-2*x))*Exp(x/(x-1)) ))); // G. C. Greubel, Mar 09 2021

a:=series(exp(x/(x - 1))*(1 - x)/(1 - 2*x), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 21; CoefficientList[Series[Exp[x/(x - 1)] (1 - x)/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Subfactorial[k] n!/k!, {k, 0, n}], {n, 0, 21}]
A317096[n_]:= A317096[n]= n!*(LaguerreL[n,1] + Sum[2^(n-j-2)*LaguerreL[j,1], {j,0, n-2}]); Table[A317096[n], {n,0,25}] (* G. C. Greubel, Mar 09 2021 *)

def A317096(n): return factorial(n)*( gen_laguerre(n,0,1) + sum(2^(n-j-2)*gen_laguerre(j,0,1) for j in (0..n-2)) )
[A317096(n) for n in (0..25)] # G. C. Greubel, Mar 09 2021

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000166(k)*n!/k!.
a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n+1). - Vaclav Kotesovec, Mar 26 2019
a(n) = n!*(LaguarreL(n,1) + Sum_{j=0..n-2} 2^(n-j-2)*LaguerreL(j,1)). - G. C. Greubel, Mar 09 2021

A323770 Expansion of e.g.f. x(2 - x)exp(x/(1 - x))/(2*(1 - x)^2).

Original entry on oeis.org

0, 1, 5, 30, 214, 1775, 16791, 178360, 2101100, 27172269, 382566025, 5823044546, 95253119490, 1666020561595, 31019392831259, 612430207741500, 12778091116288216, 280893425932078745, 6487870112636577165, 157066777096248548134, 3976727555939887035950, 105087648979005066820551

Offset: 0

Ilya Gutkovskiy, Jan 27 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..430

Cf. A000217, A052852, A103194.

[n eq 0 select 0 else (&+[Binomial(n-1,k)*Binomial(k+2,2)* Factorial(n)/Factorial(k+1): k in [0..n-1]]): n in [0..20]]; // G. C. Greubel, Mar 05 2021

seq(n!*coeff(series(x*(2-x)*exp(x/(1-x))/(2*(1-x)^2),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 29 2019

nmax = 21; CoefficientList[Series[x (2 - x) Exp[x/(1 - x)]/(2 (1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Binomial[k + 1, 2] n!/k!, {k, 0, n}], {n, 0, 21}]

[0]+[sum(binomial(n-1,k)*binomial(k+2,2)*factorial(n)/factorial(k+1) for k in (0..n-1)) for n in [1..20]] # G. C. Greubel, Mar 05 2021

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000217(k)*n!/k!.
a(n) ~ n^(n + 3/4) / (2^(3/2) * exp(n - 2*sqrt(n) + 1/2)). - Vaclav Kotesovec, Jan 27 2019
a(n) = n!*Hypergeometric2F2([1-n, 3], [1, 2], -1). - G. C. Greubel, Mar 05 2021

A347340 E.g.f.: exp( exp(exp(x) - 1) - exp(x) ).

Original entry on oeis.org

1, 0, 1, 4, 17, 91, 587, 4327, 35604, 323316, 3210600, 34574453, 400893066, 4975247460, 65755573847, 921535225267, 13643496840808, 212688569520955, 3480978391442106, 59657975022473437, 1068151956803180295, 19937983367649562025, 387243759600707804811, 7812456801157894913964

Offset: 0

Ilya Gutkovskiy, Aug 27 2021

nonn

Exponential transform of A058692.

Stirling transform of A000296.

Alois P. Heinz, Table of n, a(n) for n = 0..480

Cf. A000110, A000166, A000258, A000296, A000587, A052852, A058692, A182386, A288268.

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1), j=2..n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 27 2021
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      `if`(t=0, 1, b(j, 0)-1)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021

nmax = 23; CoefficientList[Series[Exp[Exp[Exp[x] - 1] - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (BellB[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(exp(x)-1)-exp(x)))) \\ Michel Marcus, Aug 27 2021

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * (Bell(k) - 1) * a(n-k).
a(n) = Sum_{k=0..n} Stirling2(n,k) * A000296(k).
a(n) = Sum_{k=0..n} binomial(n,k) * A000258(k) * A000587(n-k).

A358631 Irregular table T(n, k), n >= 0, k > 0, read by rows of extended (due to binary expansion of n) Stirling numbers of the first kind.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 5, 1, 6, 11, 6, 1, 6, 7, 1, 12, 20, 9, 1, 18, 26, 9, 1, 24, 50, 35, 10, 1, 8, 9, 1, 18, 29, 12, 1, 30, 41, 12, 1, 48, 94, 59, 14, 1, 36, 47, 12, 1, 72, 130, 71, 14, 1, 96, 154, 71, 14, 1, 120, 274, 225, 85, 15, 1, 10, 11, 1, 24, 38, 15, 1, 42

Offset: 0

Mikhail Kurkov, Nov 24 2022

Row n length is A000120(n) + 2.

			Irregular table begins:
    1,   1;
    2,   3,   1;
    4,   5,   1;
    6,  11,   6,  1;
    6,   7,   1;
   12,  20,   9,  1;
   18,  26,   9,  1;
   24,  50,  35, 10,  1;
    8,   9,   1;
   18,  29,  12,  1;
   30,  41,  12,  1;
   48,  94,  59, 14,  1;
   36,  47,  12,  1;
   72, 130,  71, 14,  1;
   96, 154,  71, 14,  1;
  120, 274, 225, 85, 15, 1;

Cf. A000120, A053645, A052852, A063250, A132393, A290255, A347205.

Similar tables: A358612.

b1(n)=if(n>0, my(A=n - 2^logint(n, 2)); if(A>0, logint(A, 2) + 1))
b2(n)=if(n>0, my(A=b1(3*2^logint(n, 2) - n - 1)); n + if(A>0, 2^(A-1)))
P(n,k)=if(n==0 || k==1, (n==0 && k<3) + (k==1 && n>0), my(L=logint(n, 2), A=n - 2^L); (hammingweight(A) + 2)*P(A, k-1)*(L - b1(n) + 1) + P(b2(A), k))
T(n, k)=my(A=hammingweight(n)); if(k<=(A + 2), P(n, A - k + 3))

T(n, k) = P(n, wt(n) - k + 3) for n >= 0, 0 < k <= wt(n) + 2 where wt(n) = A000120(n).
P(n, 1) = 1 for n > 0 with P(0, 1) = P(0, 2) = 1.
P(n, k) = (A000120(q(n)) + 2)*P(q(n), k-1)*(A290255(n) + 1) + P(s(q(n)), k) for n > 0, k > 1 where q(n) = A053645(n) and where s(n) = n + [A063250(n) > 0]*2^(A063250(n) - 1).
T(2^n - 1, k) = abs(Stirling1(n+2, k)) for n >= 0, k > 0.
Conjectures: (Start)
T(n, 1) = (A000120(n) + 1)!*A347205(n) for n >= 0.
Sum_{k=1..A000120(n) + 2} T(n, k)*(-1)^k = 0 for n >= 0.
Sum_{k=0..2^n - 1} Sum_{j=1..A000120(k) + 2} T(k, j) = 2*A052852(n+1) for n >= 0.
Sum_{i=1..wt(k) + 2} m^(i-1)*T(k, i) = (wt(k) + 1)!*A347205(2^m*(2k+1)) for m >= 0, k >= 0 where wt(n) = A000120(n). (End)

Offset corrected by Mikhail Kurkov, Nov 07 2024

A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 12, 6, 0, 4, 36, 72, 24, 0, 5, 80, 360, 480, 120, 0, 6, 150, 1200, 3600, 3600, 720, 0, 7, 252, 3150, 16800, 37800, 30240, 5040, 0, 8, 392, 7056, 58800, 235200, 423360, 282240, 40320, 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880

Offset: 0

Peter Luschny, Mar 28 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,   2;
  [3] 0, 3,  12,     6;
  [4] 0, 4,  36,    72,     24;
  [5] 0, 5,  80,   360,    480,     120;
  [6] 0, 6, 150,  1200,   3600,    3600,     720;
  [7] 0, 7, 252,  3150,  16800,   37800,   30240,    5040;
  [8] 0, 8, 392,  7056,  58800,  235200,  423360,  282240,   40320;
  [9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;

Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.

Cf. A144084, A021010.

A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:
seq(seq(A361893(n,k), k = 0..n), n = 0..9);
# Using the egf.:
egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):
poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):
row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);

T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).
T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).
T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).

cp's OEIS Frontend

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A300159 Number of ways of converting one set of lists containing n elements to another set of lists containing n elements by removing the last element from one of the lists and either appending it to an existing list or treating it as a new list.

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A317096 Expansion of e.g.f. ((1 - x)/(1 - 2*x))*exp(x/(x - 1)).

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