A052515 -id:A052515 - cp's OEIS frontend

A005803 Second-order Eulerian numbers: a(n) = 2^n - 2*n.

1, 0, 0, 2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072, 8166, 16356, 32738, 65504, 131038, 262108, 524250, 1048536, 2097110, 4194260, 8388562, 16777168, 33554382, 67108812, 134217674, 268435400, 536870854, 1073741764, 2147483586

Offset: 0

N. J. A. Sloane, Simon Plouffe

Starting with n=2, a(n) is the second-order Eulerian number <> (see A008517).
Also, number of 3 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (01;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004
This is the number of target DNA sequences of the correct length present after each successive cycle of the Polymerase Chain Reaction (PCR). The first two cycles produce 0 target sequences, there are 2 target sequences present after the third cycle, then 8 after the fourth cycle, and so forth. - A. Timothy Royappa, Jun 16 2012
a(n+2) = the row sums of A222403. - J. M. Bergot, Apr 04 2018

			G.f. = 1 + 2*x^3 + 8*x^4 + 22*x^5 + 52*x^6 + 114*x^7 + 240*x^8 + 494*x^9 + ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..500
S. Bilotta, E. Grazzini, and E. Pergola, Enumeration of Two Particular Sets of Minimal Permutations, J. Int. Seq. 18 (2015) 15.10.2.
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
Jim Haglund and Mirko Visontai, Stable multivariate Eulerian polynomials and generalized Stirling permutations.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Sandi Klavžar, Uroš Milutinović and Ciril Petr, Hanoi graphs and some classical numbers, Expo. Math. 23 (2005), no. 4, 371-378.
James McClung, Constructions and Applications of W-States, Bachelor Thesis, Worcester Polytechnic Institute (2020).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Sam Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv preprint arXiv:1810.00993 [math.CO], 2018 (A000246); Discrete Math, 343 (2020), article 111869.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Equivalent to second column of A008517.
a(n) = A070313 + 1 = A052515 + 2. Bisection of A077866.
Equals for n =>3 the third right hand column of A163936.
Cf. A000918 (first differences).
Cf. A000079, A000325, A005843, A130102.

a005803 n = 2 ^ n - 2 * n
a005803_list = 1 : f 1 [0, 2 ..] where
   f x (z:zs@(z':_)) = y : f y zs  where y = (x + z) * 2 - z'
-- Reinhard Zumkeller, Jan 19 2014

[2^n-2*n: n in [0..30]]; // Wesley Ivan Hurt, Jun 04 2014

A005803:=-2*z/(2*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for three leading terms

Table[2^n-2n,{n,0,50}] (* or *) LinearRecurrence[{4,-5,2},{1,0,0},51] (* Harvey P. Dale, May 21 2011 *)

{a(n) = if( n<0, 0, 2^n - 2*n)}; /* Michael Somos, Oct 13 2002 */

G.f.: 1 + 2*x^3/((1-x)^2*(1-2*x)). a(n) = A008517(n-1, 2). - Michael Somos, Oct 13 2002
Equals binomial transform of [1, -1, 1, 1, 1, ...]. - Gary W. Adamson, Jul 14 2008
a(0) = 1 and a(n) = Sum_{k=0..n-3} ((-1)^(n+k+1)*binomial(2*n-1,k)*stirling1(2*n-k-3,n-k-2)), n=>1. - Johannes W. Meijer, Oct 16 2009
a(0)=1, a(1)=0, a(2)=0, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Harvey P. Dale, May 21 2011
a(n) = A000225(n+1) - A081494(n+1), n > 1. In other words, a(n) equals the sum of the elements in a Pascal triangle of depth n+1 minus the sum of the elements on its perimeter. - Ivan N. Ianakiev, Jun 01 2014
a(n) = A165900(n-1) + Sum_{i=0..n-1} a(i), for n > 0. - Ivan N. Ianakiev, Nov 24 2014
a(n) = A000225(n) - A005408(n-1). - Miquel Cerda, Nov 25 2016
E.g.f.: exp(x)*(exp(x) - 2*x). - Ilya Gutkovskiy, Nov 25 2016

A200091 The number of ways of putting n labeled items into k labeled boxes so that each box receives at least 2 objects.

Original entry on oeis.org

1, 1, 1, 6, 1, 20, 1, 50, 90, 1, 112, 630, 1, 238, 2940, 2520, 1, 492, 11508, 30240, 1, 1002, 40950, 226800, 113400, 1, 2024, 137610, 1367520, 2079000, 1, 4070, 445896, 7271880, 22869000, 7484400, 1, 8164, 1410552, 35692800, 196396200, 194594400, 1, 16354

Offset: 2

Peter Bala, Dec 04 2011

Equivalently, the number of ordered set partitions of the set [n] into k blocks of size at least two. When the boxes are unlabeled or the set partitions unordered we obtain A008299.
The number of doubly-surjective functions f:[n]->[k], where a doubly-surjective function f has pre-image sets of size at least 2 for each element of the codomain. Also, the number of ways to distribute n different toys to k different children so that each child gets at least two toys. - Dennis P. Walsh, Apr 09 2013
T(n,k) is the number of chains 0 = x_0 < x_1 < ... < x_k = 1 in the Boolean lattice of rank n, such that x_i is not covered by x_(i+1) for all i. - Geoffrey Critzer, Jul 15 2018

			Table begins
  n |k=1   2     3     4
----+-------------------
  2 |  1
  3 |  1
  4 |  1   6
  5 |  1  20
  6 |  1  50    90
  7 |  1 112   630
  8 |  1 238  2940  2520
  9 |  1 492 11508 30240
  ...
T(4,2) = 6: The arrangements of 4 objects into 2 boxes { } and [ ] so that each box contains at least 2 items are {1,2}[3,4], {1,3}[2,4], {2,3}[1,4] and the 3 other possibilities where the contents of a pair of boxes are swapped.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, page 100-109.

Muniru A Asiru, Table of n, a(n) for n = 2..626
Marko R. Riedel, Count by PIE and symbolic combinatorics, Math Stackexchange, March 2022.
Dennis Walsh, Notes on doubly-surjective finite functions

T(n,2) = A052515(n), T(n,3) = A224541(n), T(n,4) = A224542(n).
Cf. A032032 (row sums).
Cf. A008299, A019538, A200092.

Flat(List([2..14],n->List([1..Int(n/2)],k->Sum([0..k],j->(-1)^j*Binomial(k,j)*(Sum([0..j],i->Binomial(j,i)*(Binomial(n,i)*Factorial(i)*(k-j)^(n-i)))))))); # Muniru A Asiru, Jul 17 2018

seq(seq(eval(diff((exp(x)-x-1)^k,x$n),x=0),k=1..floor(n/2)),n=2..20); # Dennis P. Walsh, Apr 09 2013
T := proc(n,k) local r; k!* add(binomial(n,r)*(-1)^r*Stirling2(n-r,k-r), r=0..min(n,k)); end; # Marko Riedel, Mar 25 2022

t[n_, k_] := k! * Sum[ (-1)^i*Binomial[n, i] * Sum[ (-1)^j*(k - i - j)^(n - i) / (j!*(k - i - j)!), {j, 0, k - i}], {i, 0, k}]; Table[ t[n, k], {n, 2, 14}, {k, 1, n/2}] // Flatten (* Jean-François Alcover, Apr 10 2013 *)

E.g.f. with additional 1: 1/(1-t*(exp(x)-1-x)) = 1 + t*x^2/2! + t*x^3/3! + (t+6*t^2)*x^4/4! + ....
E.g.f. (k fixed): (exp(x)-x-1)^k. - Dennis P. Walsh, Apr 09 2013
Recurrence relation: T(n+1,k) = k*(T(n,k) + n*T(n-1,k-1)). T(n,k) = k!*A008299(n,k).
T(n,k+j) = Sum_{i=0..n} C(n,i)*T(i,k)*T(n-i,j). - Dennis P. Walsh, Apr 09 2013
T(n,k) = Sum_{j=0..k} (-1)^j*C(k,j)*(Sum_{i=0..j} C(j,i)*C(n,i)*i!*(k-j)^(n-i)) for 1 <= k <= n/2 and n >= 2. - Dennis P. Walsh, Apr 10 2013
T(n,k) = k!*Sum_{r=0..min(n,k)} binomial(n,r)*(-1)^r*Stirling2(n-r, k-r). - Marko Riedel, Mar 25 2022

A052516 Number of pairs of sets of cardinality at least 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 20, 70, 182, 420, 912, 1914, 3938, 8008, 16172, 32526, 65262, 130764, 261800, 523906, 1048154, 2096688, 4193796, 8388054, 16776614, 33553780, 67108160, 134216970

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The number of functions f:[n]->[2] such that f maps at least 3 elements to 1 and at least 3 elements to 2. a(6) = 20 since there are C(6,3) ways to choose the 3 elements of {1,2,3,4,5,6} that f maps to 1. - Dennis P. Walsh, Dec 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 82
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A052515.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (Exp(x) -1-x-x^2/2)^2 )); [0,0,0,0,0] cat [Factorial(n+5)*b[n]: n in [1..m-6]]; // G. C. Greubel, May 13 2019

Pairs spec := [S,{S=Prod(B,B),B=Set(Z,3 <= card)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with (combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=3)}, labeled]: seq(count(subs(r=2,ZL),size=m),m=0..20); # Zerinvary Lajos, Mar 09 2007
seq(max(0,2^n-n^2-n-2), n=0..40); # Dennis P. Walsh, Dec 09 2014

Table[Max[0,2^n-n^2-n-2],{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011*)

a(n)=max(0,2^n-n^2-n-2) \\ Charles R Greathouse IV, Nov 20 2011

(2*x^6*(10-15*x+6*x^2)/((1-x)^3*(1-2*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 13 2019

E.g.f.: (exp(x) -1)^2 -x*(2+x)*exp(x) +2*x +2*x^2 +x^3 +x^4/4.
(n-1)*a(n+2) + (1-3*n)*a(n+1) + 2*(n+1)*a(n) = 0, a(0) = .. a(5) = 0, a(6) = 20.
G.f.: 2*x^6*(10-15*x+6*x^2)/((1-x)^3*(1-2*x)). - Colin Barker, Feb 19 2012
a(n) = max(0,2^n-n^2-n-2). - Dennis P. Walsh, Dec 09 2014
E.g.f.: (exp(x) - 1 - x - x^2/2)^2. - Dennis P. Walsh, Dec 09 2014

A144394 Triangle read by rows (n >= 4, 0 <= k <= n - 4): row n gives the coefficients in the expansion of ((x + 1)^n - (x^n + nx^(n - 1) + nx + 1))/x^2.

Original entry on oeis.org

6, 10, 10, 15, 20, 15, 21, 35, 35, 21, 28, 56, 70, 56, 28, 36, 84, 126, 126, 84, 36, 45, 120, 210, 252, 210, 120, 45, 55, 165, 330, 462, 462, 330, 165, 55, 66, 220, 495, 792, 924, 792, 495, 220, 66, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91

Offset: 4

Roger L. Bagula and Gary W. Adamson, Oct 02 2008

Interior of Pascal's triangle, stripping out the initial 1, n and final n, 1 in each row.

			Triangle begins:
    6;
   10,  10;
   15,  20,   15;
   21,  35,   35,   21;
   28,  56,   70,   56,   28;
   36,  84,  126,  126,   84,   36;
   45, 120,  210,  252,  210,  120,   45;
   55, 165,  330,  462,  462,  330,  165,   55;
   66, 220,  495,  792,  924,  792,  495,  220,   66;
   78, 286,  715, 1287, 1716, 1716, 1287,  715,  286,   78;
   91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001,  364,  91;
  105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105;
  ...

Reinhard Zumkeller, Rows n = 4..120 of triangle, flattened
Hermann Stamm-Wilbrandt, Sum of Pascal's triangle reciprocals [Cached copy from the Wayback Machine]
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A052515 (row sums), A024746 (sorted), A144393.

a144394 n k = a144394_tabl !! (n-4) !! k
a144394_row n = a144394_tabl !! (n-4)
a144394_tabl = map (drop 2 . reverse . drop 2) $ drop 4 a007318_tabl
-- Reinhard Zumkeller, Dec 24 2012

p[x_, n_] = ((x + 1)^n - (x^n + n*x^(n - 1) + n*x + 1))/x^2
Table[CoefficientList[p[x, n], x], {n, 4, 15}] // Flatten

create_list(binomial(n, k + 2), n, 4, 20, k, 0, n - 4); /* Franck Maminirina Ramaharo, Jan 25 2019 */

T(n,k) = binomial(n, k + 2), n >= 4, 0 <= k <= n - 4.

Sum_{n >= 4, 0 <= k <= n-4} 1/T(n,k) = 3/2. - Hermann Stamm-Wilbrandt, Jul 21 2014

Edited by Franklin T. Adams-Watters, Apr 07 2010

A224541 Number of doubly-surjective functions f:[n]->[3].

Original entry on oeis.org

90, 630, 2940, 11508, 40950, 137610, 445896, 1410552, 4390386, 13514046, 41278068, 125405532, 379557198, 1145747538, 3452182656, 10388002848, 31230066186, 93828607686, 281775226860, 845929656900, 2539047258150, 7619759016090, 22864712861880, 68605412870088

Offset: 6

Dennis P. Walsh, Apr 09 2013

Third column of A200091.
Also, a(n) is (i) the number of length-n words on the alphabet A, B, and C with each letter occurring at least twice; (ii) the number of ways to distribute n different toys to 3 different children so that each child gets at least 2 toys; (iii) the number of ways to put n numbered balls into 3 labeled boxes so that each box gets at least 2 balls; (iv) the number of n-digit positive integers consisting only of the digits 1, 2, and 3 with each of these digits appearing at least twice. A doubly-surjective function f has size at least 2 for each pre-image set, that is, |f^-1(y)|>=2 for each element y of the codomain.[Note that a surjective function has |f^-1(y)|>=1.] The triangle A200091 provides the number of doubly-surjective functions f:[n]->[k].  Column 3 of triangle A200091 is a(n).
Sequence A052515 is the number of doubly-surjective functions f:[n]->[2] with exponential generating function (exp(x)-x-1)^2. In general, the number of doubly-surjective functions f:[n]->[k] has exponential generating function (exp(x)-x-1)^k.

			For n=6 we have a(6)=90 since there are 90 six-digit  positive integers using only digits 1, 2, and 3 with each of those digits appearing at least twice. The first 30 of the ninety, namely those with initial digit 1, are given below:
112233, 112323, 112332, 113223, 113232, 113322,
121233, 121323, 121332, 122133, 122313, 122331,
123123, 123132, 123213, 123231, 123312, 123321,
131223, 131232, 131322, 132123, 132132, 132213,
132231, 132312, 132321, 133122, 133212, 133221.

Dennis Walsh, Notes on doubly-surjective finite functions

Cf. A052515, the number of doubly-surjective functions f:[n]->[2].

seq(3^n-3*2^n-3*n*2^(n-1)+3+3*n+3*n^2, n=6..40);

With[{nn=40},Drop[CoefficientList[Series[(Exp[x]-x-1)^3,{x,0,nn}],x] Range[0,nn]!,6]] (* Harvey P. Dale, Oct 01 2015 *)

x='x+O('x^66); Vec(serlaplace((exp(x)-x-1)^3)) \\ Joerg Arndt, Apr 10 2013

a(n) = 3^n-3*2^n-3*n*2^(n-1)+3+3*n+3*n^2.
E.g.f.: (exp(x)-x-1)^3.
From Alois P. Heinz, Apr 10 2013: (Start)
a(n) = 6*A000478(n).
G.f.: -6*(12*x^3-40*x^2+45*x-15)*x^6 / ((3*x-1)*(2*x-1)^2*(x-1)^3).
(End)

A322291 Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).

Original entry on oeis.org

1, 2, 3, 3, 6, 7, 6, 10, 14, 15, 10, 20, 25, 30, 31, 20, 35, 50, 56, 62, 63, 35, 70, 91, 112, 119, 126, 127, 70, 126, 182, 210, 238, 246, 254, 255, 126, 252, 336, 420, 456, 492, 501, 510, 511, 252, 462, 672, 792, 912, 957, 1002, 1012, 1022, 1023, 462, 924, 1254, 1584, 1749, 1914, 1969, 2024, 2035, 2046, 2047

Offset: 1

Stefano Spezia, Aug 28 2019

T(n, k) is a sharp upper bound on the cardinality of a k-antichain in {0, 1}^n due to P. Erdős.

T(n, k) is also the total number of compositions with first part k, n+1 parts, and all differences between adjacent parts in {-1,1}. - John Tyler Rascoe, May 07 2023

			n\k|   1    2    3    4    5    6
---+-----------------------------
1  |   1
2  |   2    3
3  |   3    6    7
4  |   6   10   14   15
5  |  10   20   25   30   31
6  |  20   35   50   56   62   63
...

John Tyler Rascoe, Rows n = 1..141 of the triangle, flattened
P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51 (1945), 898-902.
C. Pelekis and V. Vlasák, On k-antichains in the unit n-cube, arXiv:1908.04727 [math.CA], 2019.

Cf. A000225 (diagonal), A189390 (row sums).

Cf. A000247, A000918, A001405, A006126, A007318, A052515, A052516, A263857, A272352, A306550.

Flat(List([1..11], n->List([1..n], k->Sum([1..k], i->Binomial(n, Int((n-k)/2)+i)))));

a:=(n, k)->sum(binomial(n, floor((1/2)*n-(1/2)*k)+i), i = 1..k): seq(seq(a(n, k), k = 1..n), n = 1..11);

T[n_,k_]:=Sum[Binomial[n,Floor[(n-k)/2]+i],{i,1,k}]; Table[T[n,k],{n,1,11},{k,1,n}]

T(n, k) = sum(i=1, k, binomial(n, floor((n-k)/2)+i));

T(n, n) = A000225(n).
T(n, n-1) = A000918(n).
T(n, n-2) = A000247(n).
T(n, n-3) = A052515(n).
T(n, n-4) = A272352(n+1).
T(n, n-5) = A052516(n).

A352607 Triangle read by rows. T(n, k) = Bell(k)Sum_{j=0..k}(-1)^(k+j)binomial(n, n-k+j)*Stirling2(n-k+j, j) for n >= 0 and 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 6, 0, 1, 20, 0, 1, 50, 75, 0, 1, 112, 525, 0, 1, 238, 2450, 1575, 0, 1, 492, 9590, 18900, 0, 1, 1002, 34125, 141750, 49140, 0, 1, 2024, 114675, 854700, 900900, 0, 1, 4070, 371580, 4544925, 9909900, 2110185

Offset: 0

Peter Luschny and Mélika Tebni, Mar 23 2022

			Triangle starts:
[0] 1;
[1] 0;
[2] 0, 1;
[3] 0, 1;
[4] 0, 1,   6;
[5] 0, 1,  20;
[6] 0, 1,  50,   75;
[7] 0, 1, 112,  525;
[8] 0, 1, 238, 2450,  1575;
[9] 0, 1, 492, 9590, 18900;

Cf. A028248 (row sums), A052515 (column 2), A081066, A008299, A000110, A137375.

A352607 := (n, k) -> combinat:-bell(k)*add((-1)^(k+j)*binomial(n, n-k+j)* Stirling2(n-k+j, j), j = 0..k):
seq(seq(A352607(n, k), k = 0..n/2), n = 0..12);
# Second program:
egf := k -> combinat[bell](k)*(exp(x) - 1 - x)^k/k!:
A352607 := (n, k) -> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A352607(n, k), k = 0..n/2)), n=0..12);
# Recurrence:
A352607 := proc(n, k) option remember;
if k > n/2 then 0 elif k = 0 then k^n else k*A352607(n-1, k) +
combinat[bell](k)/combinat[bell](k-1)*(n-1)*A352607(n-2, k-1) fi end:
seq(print(seq(A352607(n, k), k=0..n/2)), n=0..12); # Mélika Tebni, Mar 24 2022

T[n_, k_] := BellB[k]*Sum[(-1)^(k+j)*Binomial[n, n-k+j]*StirlingS2[n-k+j, j], {j, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Oct 21 2023 *)

T(n, k) = (-1)^k*A000110(k)*A137375(n, k) = A000110(k)*A008299(n, k).
T(2*n, n) = A081066(n).
E.g.f. column k: Bell(k)*(exp(x) - 1 - x)^k / k!, k >= 0.
T(n, k) = Bell(k)*Sum_{j=0..k} Sum_{i=0..j} ((-1)^j*(k-j)^(n-i)*binomial(n, i)) / ((k - j)!*(j - i)!).

A224542 Number of doubly-surjective functions f:[n]->[4].

Original entry on oeis.org

2520, 30240, 226800, 1367520, 7271880, 35692800, 165957792, 742822080, 3234711480, 13803744864, 58021888080, 241116750624, 993313349544, 4064913201216, 16549636147968, 67112688842496, 271323921459096, 1094303232174240, 4405390451382960, 17709538489849440

Offset: 8

Dennis P. Walsh, Apr 09 2013

Fourth column of A200091: A200091(n,4)=a(n).
Also, a(n) is (i) the number of length-n words on the alphabet A, B, C, and D with each letter of the alphabet occurring at least twice; (ii) the number of ways to distribute n different toys to 4 children so that each child gets at least two toys; (iii) the number of ways to put n numbered balls into 4 labeled boxes so that each box gets at least two balls; (iv) the number of n-digit positive integers consisting of only digits 1,2,3, and 4 with each of these digits appearing at least twice.
A doubly-surjective function f:D->C is such that the pre-image set f^-1(y) has size at least 2 for each y in C.
Triangle A200091 provides the number of doubly-surjective functions f from a set of size n onto a set of size k. Hence a(n) is column 4 of A200091.

			a(9) = 30240 since there are 30240 ways to distribute 9 different toys to 4 children so that each child gets at least 2 toys. One child must get 3 toys and the other children get 2 toys each. There are 4 ways to pick the lucky kid. There are C(9,3) ways to choose the 3 toys for the lucky kid. There are 6!/(2!)^3 ways to distribute the remaining 6 toys among the 3 kids. We obtain 4*C(9,3)*6!/8=30240.

Dennis Walsh, Notes on doubly-surjective finite functions

Cf. A224541, A052515.

seq(eval(diff((exp(x)-x-1)^4,x$n),x=0),n=8..40);

nn=27; Drop[Range[0,nn]! CoefficientList[Series[(Exp[x]-x-1)^4, {x,0,nn}], x], 8] (* Geoffrey Critzer, Sep 28 2013 *)

my(x='x+O('x^66)); Vec(serlaplace((exp(x)-x-1)^4)) /* Joerg Arndt, Apr 10 2013 */

a(n) = 4^n-4*3^n-4*n*3^(n-1)+(9*n+3*n^2)*2^(n-1)+6*2^n-4-8*n-4*n^3;
a(n) = sum(n!/(i!*j!*k!*m!)) over  such that i,j,k, and m are all at least 2 and i+j+k+m=n.
E.g.f.: (exp(x)-x-1)^4.
a(n) = 24*A058844(n). - Alois P. Heinz, Apr 10 2013
G.f.: 24*x^8*(288*x^6-1560*x^5+3500*x^4-4130*x^3+2625*x^2-840*x+105) / ((x-1)^4*(2*x-1)^3*(3*x-1)^2*(4*x-1)). - Colin Barker, Jun 04 2013

A375659 For 0<=k<=n, T(n,k) = the number of Dyck-type lattice paths of length n, starting at the point (0,k), triangle T read by rows.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 6, 7, 8, 6, 10, 14, 15, 16, 10, 20, 25, 30, 31, 32, 20, 35, 50, 56, 62, 63, 64, 35, 70, 91, 112, 119, 126, 127, 128, 70, 126, 182, 210, 238, 246, 254, 255, 256, 126, 252, 336, 420, 456, 492, 501, 510, 511, 512, 252, 462, 672, 792, 912, 957, 1002, 1012, 1022, 1023, 1024

Offset: 0

Marilena Jianu, Aug 23 2024

A Dyck type lattice path has the steps (1,1) or (1,-1) and never passes below the x-axis.
For k>=n, the number of Dyck-type lattice paths is 2^n.
The sequence completes A322291 by adding a diagonal of powers of 2.

			  n | k=0    1    2    3    4    5    6    7
 ---+---------------------------------------
  0 |  1
  1 |  1    2
  2 |  2    3    4
  3 |  3    6    7    8
  4 |  6   10   14   15   16
  5 | 10   20   25   30   31   32
  6 | 20   35   50   56   62   63   64
  7 | 35   70   92  112  119  126  127  128

John Tyler Rascoe, Rows n = 0..140, flattened

T(n,0) = T(n-1,1) = A001405(n).
T(n,n) = A000079(n).
T(n,n-1) = A000225(n).
T(n,n-2) = A000918(n).
T(n,n-3) = A000247(n).
T(n,n-4) = A052515(n).
Row sums = A189162(n+1).
Cf. A322291, A368175.

a:=(n,k)->sum(binomial(n, floor((1/2)*(n-k))+i), i = 0..k):
seq(seq(a(n, k), k = 0..n), n = 0..11);

from math import comb
def A375659(n,k):
    return sum(comb(n,i+(n-k)//2) for i in range(k+1)) # John Tyler Rascoe, Sep 04 2024

T(n,k) = Sum_{i = 0..k} binomial(n, floor((n-k)/2)+i).

T(n,k) = T(n-1,k-1)+T(n-1,k+1), for all n>=2 and 1<=k<=n-2.

A308804 Triangular table of coefficients of p in p^(k+2)/(1-p) LerchPhi(1-p,-1-k,(p-1)/p) as function of k=1..n.

Original entry on oeis.org

1, 2, -1, 9, -9, 1, 44, -66, 24, -1, 265, -530, 320, -55, 1, 1854, -4635, 3940, -1275, 118, -1, 14833, -44499, 48825, -23485, 4571, -245, 1, 133496, -467236, 628544, -403270, 123368, -15400, 500, -1, 1334961, -5339844, 8510376, -6841674, 2885694, -598416, 49914, -1011, 1, 14684570, -66080565, 121759560, -117782490, 63630588, -18808230, 2752320, -157785, 2034, -1

Offset: 1

Wouter Meeussen, Jun 25 2019

Relations to other sequences are tentative (checked up to 24 rows). Related to the central moments of a geometric probability distribution.

Each row sums to 1. Row sums of absolute values are A091346. First column is A000166. First moments appear to be A052515 with shifted index.

Wolfram Language & System Documentation Center, LerchPhi (then click in Applications).

Cf. A091346, A000166, A052515.

Table[CoefficientList[ p^(k + 2)/(1 - p) LerchPhi[1 - p, -k - 1, (-1 + p)/p], p], {k, 1, 12}]

cp's OEIS Frontend

A005803 Second-order Eulerian numbers: a(n) = 2^n - 2*n.

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A200091 The number of ways of putting n labeled items into k labeled boxes so that each box receives at least 2 objects.

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A052516 Number of pairs of sets of cardinality at least 3.

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A144394 Triangle read by rows (n >= 4, 0 <= k <= n - 4): row n gives the coefficients in the expansion of ((x + 1)^n - (x^n + n*x^(n - 1) + n*x + 1))/x^2.

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A224541 Number of doubly-surjective functions f:[n]->[3].

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A322291 Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).

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A144394 Triangle read by rows (n >= 4, 0 <= k <= n - 4): row n gives the coefficients in the expansion of ((x + 1)^n - (x^n + nx^(n - 1) + nx + 1))/x^2.

A352607 Triangle read by rows. T(n, k) = Bell(k)Sum_{j=0..k}(-1)^(k+j)binomial(n, n-k+j)*Stirling2(n-k+j, j) for n >= 0 and 0 <= k <= floor(n/2).