A005564 -id:A005564 - cp's OEIS frontend

A001701 Generalized Stirling numbers.

1, 6, 26, 71, 155, 295, 511, 826, 1266, 1860, 2640, 3641, 4901, 6461, 8365, 10660, 13396, 16626, 20406, 24795, 29855, 35651, 42251, 49726, 58150, 67600, 78156, 89901, 102921, 117305, 133145, 150536, 169576, 190366, 213010, 237615, 264291, 293151, 324311

Offset: 1

N. J. A. Sloane

For n>3, a(n-2) gives the number of bounded regions created when the pairwise perpendicular bisectors of n points divide the Euclidean plane into a maximum of A308305(n) regions. This is also equivalent to the number of regions lost from A308305(n) when n>3 points move from maximal position to a circle. - Alvaro Carbonero, Elizabeth Castellano, Charles Kulick, Karie Schmitz, Jul 26 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 1..1000
Alvaro Carbonero, Beth Anne Castellano, Gary Gordon, Charles Kulick, Karie Schmitz, and Brittany Shelton, Permutations of point sets in R_d, arXiv:2106.14140 [math.CO], 2021.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
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
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Equals A059302(n+2) + 1, n>1. Partial sums of A005564.

For n>1, a(n) = A145324(n+1,3).

Concatenation([1],List([2..40],n->n*(n-1)*(3*n^2+17*n+26)/24)); # Muniru A Asiru, Sep 29 2018

[1] cat [n*(n-1)*(3*n^2 + 17*n + 26)/24: n in [2..40]]; // Vincenzo Librandi, Sep 30 2018

A001701 := proc(n)
    if n = 1 then
        1;
    else
        n*(n-1)*(3*n^2+17*n+26)/24 ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016

f[k_] := k + 1; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[2, t[n]]; Join[{1}, Table[a[n], {n, 2, 30}]] (* Clark Kimberling, Dec 31 2011 *)
Join[{1}, Table[n (n - 1) (3 n^2 + 17 n + 26) / 24, {n, 2, 40}]] (* Vincenzo Librandi, Sep 30 2018 *)
CoefficientList[Series[(-1 - x - 6 x^2 + 9 x^3 - 5 x^4 + x^5)/(-1 + x)^5, {x, 0, 30}], x] (* Stefano Spezia, Sep 30 2018 *)
Prepend[Table[Coefficient[Product[x+j, {j,2,k}], x, k-3], {k,3,40}],1] (* or *) Prepend[LinearRecurrence[{5, -10, 10, -5, 1}, {6, 26, 71, 155, 295}, 40],1] (*Robert A. Russell, Oct 04 2018 *)

Vec(x*(-1-x-6*x^2+9*x^3-5*x^4+x^5)/(-1+x)^5+O(x^30)) \\ Stefano Spezia, Sep 30 2018

a(n) = n*(n-1)*(3n^2 + 17n + 26)/24, n > 1.
G.f.: z*(-1-z-6*z^2+9*z^3-5*z^4+z^5)/(z-1)^5. - Simon Plouffe in his 1992 dissertation
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i) * Product_{j=0..k-1} (-a - j), then a(n) = f(n,n-2,2), for n >= 2. - Milan Janjic, Dec 20 2008
For n>1, a(n) = A308305(n+2) - (n^2 + 3n + 2). - Alvaro Carbonero, Elizabeth Castellano, Charles Kulick, Karie Schmitz, Jul 20 2019
E.g.f.: x + (1/24)*exp(x)*x^2*(72 + 32*x + 3*x^2). - Stefano Spezia, Sep 07 2019
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Colin Barker, Jul 08 2020

A288318 Number T(n,k) of Dyck paths of semilength n such that each positive level has exactly k peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 4, 3, 0, 0, 1, 0, 6, 6, 0, 0, 0, 1, 0, 8, 0, 4, 0, 0, 0, 1, 0, 24, 9, 20, 0, 0, 0, 0, 1, 0, 52, 54, 20, 5, 0, 0, 0, 0, 1, 0, 96, 138, 0, 45, 0, 0, 0, 0, 0, 1, 0, 212, 207, 16, 105, 6, 0, 0, 0, 0, 0, 1, 0, 504, 360, 200, 70, 84, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Jun 07 2017

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms for k<=n. T(0,k) = 1 and T(n,k) = 0 if k > n > 0.

			. T(5,1) = 4:
.               /\        /\          /\        /\
.            /\/  \      /  \/\    /\/  \      /  \/\
.         /\/      \  /\/      \  /      \/\  /      \/\ .
.
. T(5,2) = 3:
.              /\/\      /\/\      /\/\
.         /\/\/    \  /\/    \/\  /    \/\/\  .
.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  0,  1;
  0,  2,  0,  1;
  0,  0,  0,  0, 1;
  0,  4,  3,  0, 0, 1;
  0,  6,  6,  0, 0, 0, 1;
  0,  8,  0,  4, 0, 0, 0, 1;
  0, 24,  9, 20, 0, 0, 0, 0, 1;
  0, 52, 54, 20, 5, 0, 0, 0, 0, 1;

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Counting lattice paths

Columns k=0-10 give: A000007, A287846, A287845, A288319, A288320, A288321, A288322, A288323, A288324, A288325, A288326.
Row sums give A287987.
Cf. A000108, A000984, A005564, A288108, A288940.

b:= proc(n, k, j) option remember;
     `if`(n=j, 1, add(b(n-j, k, i)*(binomial(i, k)
      *binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
    end:
T:= (n, k)-> `if`(n=0, 1, b(n, k$2)):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[b[n - j, k, i]*(Binomial[i, k]*Binomial[j - 1, i - 1 - k]), {i, 1, Min[j + k, n - j]}]];
T[n_, k_] := If[n == 0, 1, b[n, k, k]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)

T(n,n) = 1.
T(n+1,n) = 0.
T(2*n+1,n) = (n+1) for n>0.
T(2*n+2,n) = A005564(n+1) for n>1.
T(3*n,n) = A000984(n) = binomial(2*n,n).
T(3*n+1,n) = 0.
T(3*n+2,n) = (n+1)^2 for n>0.

A077414 a(n) = n(n - 1)(n + 2)/2.

Original entry on oeis.org

0, 4, 15, 36, 70, 120, 189, 280, 396, 540, 715, 924, 1170, 1456, 1785, 2160, 2584, 3060, 3591, 4180, 4830, 5544, 6325, 7176, 8100, 9100, 10179, 11340, 12586, 13920, 15345, 16864, 18480, 20196, 22015, 23940, 25974, 28120, 30381, 32760, 35260

Offset: 1

Wolfdieter Lang, Nov 29 2002

Number of independent components of a certain 3-tensor in n-space.
a(n) is the number of independent components of a 3-tensor t(a,b,c) which satisfies t(a,b,c) = t(b,a,c) and Sum_{a=1..n} t(a,a,c) = 0 for all c, with a,b,c range 1..n. (3-tensor in n-dimensional space which is symmetric and traceless in one pair of its indices.)
Row 2 of the convolution array A213761.  - Clark Kimberling, Jul 04 2012
Also, the number of ways to place two dominoes horizontally in the same row on an (n+2) X (n+2) chessboard. - Ralf Stephan, Jun 09 2014
Also, the sum of all the numbers in a completely filled n X n tic-tac-toe board with n-1 players using the numbers 0, 1, 2,... n-2. See "Sums of Square Tic Tac Toe Boards that end in a Draw" in links for proof. - Tanner Robeson, Aug 23 2020
a(n) is the number of degrees of freedom in a tetrahedral cell for a Raviart-Thomas finite element space of order n. - Matthew Scroggs, Jan 02 2021

			For n=6, a(6) = 1*(3*5+1)+2*(3*4+1)+3*(3*3+1)+4*(3*2+1)+5*(3*1+1) = 120. - _Bruno Berselli_, Feb 13 2014
G.f. = 4*x^2 + 15*x^3 + 36*x^4 + 70*x^5 + 120*x^6 + 189*x^7 + 280*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
DefElement, Raviart-Thomas
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Tanner Robeson, Sums of Square Tic Tac Toe Boards that end in a Draw.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index to sequences related to polygonal numbers.

Cf. A000096, A005564, A057145, A115067 (first differences), A213761.

Cf. similar sequences of the type m*(m+1)*(m+k)/2 listed in A267370.

[n*(n-1)*(n+2)/2: n in [1..30]]; // G. C. Greubel, Jan 18 2018

A077414:=n->n*(n-1)*(n+2)/2: seq(A077414(n), n=1..60); # Wesley Ivan Hurt, Apr 09 2017

Table[(n (n - 1) (n + 2))/2, {n, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 4, 15, 36}, 50] (* Harvey P. Dale, Jun 04 2012 *)
CoefficientList[Series[x (4 - x)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2014 *)

a(n)=n*(n-1)*(n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

concat(0, Vec(x^2*(4-x)/(1-x)^4 + O(x^200))) \\ Altug Alkan, Jan 15 2016

a(n) = n * ( binomial(n+1, 2)-1 ).
G.f.: x^2*(4-x)/(1-x)^4.
a(n) = n*Sum_{j=2..n} j. - Zerinvary Lajos, Sep 12 2006
a(1)=0, a(2)=4, a(3)=15, a(4)=36; for n>4, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Jun 04 2012
a(n) = Sum_{i=1..n-1} i*(3*(n-i)+1). - Bruno Berselli, Feb 13 2014
a(-n) = -A005564(n). - Michael Somos, Jun 09 2014
a(n) = A057145(n,n+2). - R. J. Mathar, Jul 28 2016
a(n) = t(n,t(n,1)) + n, where t(n,k) = n*(n+1)/2 + k*n and t(n,1) = A000096(n). - Bruno Berselli, Feb 28 2017
a(n) = n^3/2 + n^2/2 - n. - Tanner Robeson, Aug 23 2020
Sum_{n>=2} 1/a(n) = 7/18. - Amiram Eldar, Oct 07 2020
Sum_{n>=2} (-1)^n/a(n) = 4*log(2)/3 - 13/18. - Amiram Eldar, Feb 22 2022
E.g.f.: exp(x)*x^2*(4 + x)/2. - Stefano Spezia, Jan 03 2023

A055137 Regard triangle of rencontres numbers (see A008290) as infinite matrix, compute inverse, read by rows.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -2, -3, 0, 1, -3, -8, -6, 0, 1, -4, -15, -20, -10, 0, 1, -5, -24, -45, -40, -15, 0, 1, -6, -35, -84, -105, -70, -21, 0, 1, -7, -48, -140, -224, -210, -112, -28, 0, 1, -8, -63, -216, -420, -504, -378, -168, -36, 0, 1, -9, -80, -315, -720

Offset: 0

Christian G. Bower, Apr 25 2000

The n-th row consists of coefficients of the characteristic polynomial of the adjacency matrix of the complete n-graph.
Triangle of coefficients of det(M(n)) where M(n) is the n X n matrix m(i,j)=x if i=j, m(i,j)=i/j otherwise. - Benoit Cloitre, Feb 01 2003
T is an example of the group of matrices outlined in the table in A132382--the associated matrix for rB(0,1). The e.g.f. for the row polynomials is exp(x*t) * exp(x) *(1-x). T(n,k) = Binomial(n,k)* s(n-k) where s = (1,0,-1,-2,-3,...) with an e.g.f. of exp(x)*(1-x) which is the reciprocal of the e.g.f. of A000166. - Tom Copeland, Sep 10 2008
Row sums are: {1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}. - Roger L. Bagula, Feb 20 2009
T is related to an operational calculus connecting an infinitesimal generator for fractional integro-derivatives with the values of the Riemann zeta function at positive integers (see MathOverflow links). - Tom Copeland, Nov 02 2012
The submatrix below the null subdiagonal is signed and row reversed A127717. The submatrix below the diagonal is A074909(n,k)s(n-k) where s(n)= -n, i.e., multiply the n-th diagonal by -n. A074909 and its reverse A135278 have several combinatorial interpretations. - Tom Copeland, Nov 04 2012
T(n,k) is the difference between the number of even (A145224) and odd (A145225) permutations (of an n-set) with exactly k fixed points. - Julian Hatfield Iacoponi, Aug 08 2024

			1; 0,1; -1,0,1; -2,-3,0,1; -3,-8,-6,0,1; ...
(Bagula's matrix has a different sign convention from the list.)
From _Roger L. Bagula_, Feb 20 2009: (Start)
  { 1},
  { 0,   1},
  {-1,   0,    1},
  { 2,  -3,    0,    1},
  {-3,   8,   -6,    0,     1},
  { 4, -15,   20,  -10,     0,    1},
  {-5,  24,  -45,   40,   -15,    0,    1},
  { 6, -35,   84, -105,    70,  -21,    0,   1},
  {-7,  48, -140,  224,  -210,  112,  -28,   0,   1},
  { 8, -63,  216, -420,   504, -378,  168, -36,   0, 1},
  {-9,  80, -315,  720, -1050, 1008, -630, 240, -45, 0, 1}
(End)
R(3,x) = (-1)^3*Sum_{permutations p in S_3} sign(p)*(-x)^(fix(p)).
    p   | fix(p) | sign(p) | (-1)^3*sign(p)*(-x)^fix(p)
========+========+=========+===========================
  (123) |    3   |    +1   |      x^3
  (132) |    1   |    -1   |       -x
  (213) |    1   |    -1   |       -x
  (231) |    0   |    +1   |       -1
  (312) |    0   |    +1   |       -1
  (321) |    1   |    -1   |       -x
========+========+=========+===========================
                           | R(3,x) = x^3 - 3*x - 2
- _Peter Bala_, Aug 08 2011

Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993. p. 17.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p.184 problem 3.

Problem B6, The 66th William Lowell Putnam Mathematical Competition Saturday, Dec 03 2005
M. Bhargava, K. Kedlaya, and L. Ng, Solutions to the 66th William Lowell Putnam Mathematical Competition Saturday, Dec 03 2005
T. Copeland, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants
T. Copeland, Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

Cf. A005563, A005564 (absolute values of columns 1, 2).
Cf. A008290, A133314, A238363, A238385.
Cf. A000312.
Cf. A145224, A145225, A003221, A000387.

M[n_] := Table[If[i == j, x, 1], {i, 1, n}, {j, 1, n}]; a = Join[{{1}}, Flatten[Table[CoefficientList[Det[M[n]], x], {n, 1, 10}]]] (* Roger L. Bagula, Feb 20 2009 *)
t[n_, k_] := (k-n+1)*Binomial[n, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2013, after Pari *)

T(n,k)=(1-n+k)*if(k<0 || k>n,0,n!/k!/(n-k)!)

G.f.: (x-n+1)*(x+1)^(n-1) = Sum_(k=0..n) T(n,k) x^k.
T(n, k) = (1-n+k)*binomial(n, k).
k-th column has o.g.f. x^k(1-(k+2)x)/(1-x)^(k+2). k-th row gives coefficients of (x-k)(x+1)^k. - Paul Barry, Jan 25 2004
T(n,k) = Coefficientslist[Det[Table[If[i == j, x, 1], {i, 1, n}, {k, 1, n}],x]. - Roger L. Bagula, Feb 20 2009
From Peter Bala, Aug 08 2011: (Start)
Given a permutation p belonging to the symmetric group S_n, let fix(p) be the number of fixed points of p and sign(p) its parity. The row polynomials R(n,x) have a combinatorial interpretation as R(n,x) = (-1)^n*Sum_{permutations p in S_n} sign(p)*(-x)^(fix(p)). An example is given below.
Note: The polynomials P(n,x) = Sum_{permutations p in S_n} x^(fix(p)) are the row polynomials of the rencontres numbers A008290. The integral results Integral_{x = 0..n} R(n,x) dx = n/(n+1) = Integral_{x = 0..-1} R(n,x) dx lead to the identities Sum_{p in S_n} sign(p)*(-n)^(1 + fix(p))/(1 + fix(p)) = (-1)^(n+1)*n/(n+1) = Sum_{p in S_n} sign(p)/(1 + fix(p)). The latter equality was Problem B6 in the 66th William Lowell Putnam Mathematical Competition 2005. (End)
From Tom Copeland, Jul 26 2017: (Start)
The e.g.f. in Copeland's 2008 comment implies this entry is an Appell sequence of polynomials P(n,x) with lowering and raising operators L = d/dx and R = x + d/dL log[exp(L)(1-L)] = x+1 - 1/(1-L) = x - L - L^2 - ... such that L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x).
P(n,x) = (1-L) exp(L) x^n = (1-L) (x+1)^n = (x+1)^n - n (x+1)^(n-1) = (x+1-n)(x+1)^(n-1) = (x+s.)^n umbrally, where (s.)^n = s_n = P(n,0).
The formalism of A133314 applies to the pair of entries A008290 and A055137.
The polynomials of this pair P_n(x) and Q_n(x) are umbral compositional inverses; i.e., P_n(Q.(x)) = x^n = Q_n(P.(x)), where, e.g., (Q.(x))^n = Q_n(x).
The exponential infinitesimal generator (infinigen) of this entry is the negated infinigen of A008290, the matrix (M) noted by Bala, related to A238363. Then e^M = [the lower triangular A008290], and e^(-M) = [the lower triangular A055137]. For more on the infinigens, see A238385. (End)
From the row g.f.s corresponding to Bagula's matrix example below, the n-th row polynomial has a zero of multiplicity n-1 at x = 1 and a zero at x = -n+1. Since this is an Appell sequence dP_n(x)/dx = n P_{n-1}(x), the critical points of P_n(x) have the same abscissas as the zeros of P_{n-1}(x); therefore, x = 1 is an inflection point for the polynomials of degree > 2 with P_n(1) = 0, and the one local extremum of P_n has the abscissa x = -n + 2 with the value (-n+1)^{n-1}, signed values of A000312. - Tom Copeland, Nov 15 2019
From Julian Hatfield Iacoponi, Aug 08 2024: (Start)
T(n,k) = A145224(n,k) - A145225(n,k).
T(n,k) = binomial(n,k)*(A003221(n-k)-A000387(n-k)). (End)

Additional comments from Michael Somos, Jul 04 2002

A093768 Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 8, 6, 1, 4, 15, 20, 20, 1, 5, 24, 45, 75, 50, 1, 6, 35, 84, 189, 210, 175, 1, 7, 48, 140, 392, 588, 784, 490, 1, 8, 63, 216, 720, 1344, 2352, 2352, 1764, 1, 9, 80, 315, 1215, 2700, 5760, 7560, 8820, 5292, 1, 10, 99, 440, 1925, 4950, 12375, 19800

Offset: 0

Paul D. Hanna, Apr 16 2004

Suggested by Bozydar Dubalski (slawb(AT)atr.bydgoszcz.pl). Related to walks on a square lattice: main diagonal forms A005558, secondary diagonals form A005559, A005560, A005561, A005562, A005563.

Apparently row-reversed version of A052174. - R. J. Mathar, Feb 03 2025

			1;
1, 1;
1, 2, 3;
1, 3, 8, 6;
1, 4, 15, 20, 20;
1, 5, 24, 45, 75, 50;
1, 6, 35, 84, 189, 210, 175;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A088459, A005558-A005562, A005563 (column 3), A005564 (column 4), A005565 (column 5), A005566 (row sums).

A093768 := proc(n,k)
    if k = 0 then
        A088459(n,k);
    else
        A088459(n,k)-A088459(n,k-1);
    end if;
end proc:
seq(seq(A093768(n,k),k=0..n-1),n=1..10) ; # R. J. Mathar, Apr 02 2017

T[n_, k_] := Binomial[n + 1, Ceiling[k/2]]*Binomial[n, Floor[k/2]] - Binomial[n + 1, Ceiling[(k - 1)/2]]*Binomial[n, Floor[(k - 1)/2]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 25 2017 *)

{T(n,k) =binomial(n+1,ceil(k/2))*binomial(n,floor(k/2)) -binomial(n+1,ceil((k-1)/2))*binomial(n,floor((k-1)/2))}

T(n, k) = C(n+1, ceiling(k/2))*C(n, floor(k/2)) - C(n+1, ceiling((k-1)/2))*C(n, floor((k-1)/2)) for n>=k>=0.

A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

Original entry on oeis.org

1, 1, 6, 22, 55, 111, 196, 316, 477, 685, 946, 1266, 1651, 2107, 2640, 3256, 3961, 4761, 5662, 6670, 7791, 9031, 10396, 11892, 13525, 15301, 17226, 19306, 21547, 23955, 26536, 29296, 32241, 35377, 38710, 42246, 45991, 49951, 54132, 58540, 63181

Offset: 0

Paul Barry, Mar 21 2003

Diagonal in array of n-gonal numbers A081422.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005564, A081422, A081436, A081437, A081438.

List([0..40], n-> (2*n^3-n^2-n+2)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3-n^2-n+2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

a:= n-> (2*n^3-n^2-n+2)/2: seq(a(n), n=0..50); # Zerinvary Lajos, Sep 13 2006

Table[(2n^3-n^2-n+2)/2,{n,0,40}] (* Harvey P. Dale, May 29 2012 *)
CoefficientList[Series[(1 - 4 x + 11 x^2 - 8 x^3) / (1 - x)^5, {x, 0, 50}],x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (2*n^3-n^2-n+2)/2) \\ G. C. Greubel, Aug 14 2019

[(2*n^3-n^2-n+2)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

G.f.: (1 -4*x +11*x^2 -8*x^3)/(1-x)^5.
a(n) = (n + 1)*(2*n^2 - 3*n + 2)/2 = (n-1)*A005564(n+1) - n*A005564(n), where A005564(0..2) = 0, -1, 0. - Bruno Berselli, May 19 2015
E.g.f.: (2 + 5*x^2 + 2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A193002 Triangle T(n,k)=0 (k odd), T(0,0)=-3, T(n,0)=1 (n > 0) and T(n,k) = T(n-1,k) - T(n-2,k-2).

Original entry on oeis.org

-3, 1, 0, 1, 0, 3, 1, 0, 2, 0, 1, 0, 1, 0, -3, 1, 0, 0, 0, -5, 0, 1, 0, -1, 0, -6, 0, 3, 1, 0, -2, 0, -6, 0, 8, 0, 1, 0, -3, 0, -5, 0, 14, 0, -3, 1, 0, -4, 0, -3, 0, 20, 0, -11, 0, 1, 0, -5, 0, 0, 0, 25, 0, -25, 0, 3, 1, 0, -6

Offset: 0

Paul Curtz, Jul 14 2011

Consider an array with recurrence BB(m,n) = BB(m,n-1) + BB(m-1,n), m >= 0:
  3, -1, -1, -1,  -1,  -1,  -1,  -1,  -1,  -1,   -1,
  3,  2,  1,  0,  -1,  -2,  -3,  -4,  -5,  -6,   -7,
  3,  5,  6,  6,   5,   3,   0,  -4,  -9, -15,  -22,
  3,  8, 14, 20,  25,  28,  28,  24,  15,   0,  -22,
  3, 11, 25, 45,  70,  98, 126, 150, 165, 165,  143,
  3, 14, 39, 84, 154, 252, 378, 528, 693, 858, 1001,
with BB(m,n) = (3m-n)*binomial(n+m-1,n)/m if m > 0. So the BB are polynomials of degree m in n:
BB(1,n) = -(n-3)/1,
BB(2,n) = -(n-6)*(n+1)/2,    (see A055999)
BB(3,n) = -(n-9)*(n+1)*(n+2)/6,
BB(4,n) = -(n-12)*(n+1)*(n+2)*(n+3)/24,
BB(5,n) = -(n-15)*(n+1)*(n+2)*(n+3)*(n+4)/120.
Columns in the array are A010701, A016789, A095794, A005564, A059302.
T(n,k) is a zero-padded, column-shifted, sign-modified transpose of this array.

			Triangle begins
  -3;
   1,   0;
   1,   0,   3;
   1,   0,   2,   0;
   1,   0,   1,   0,  -3;
   1,   0,   0,   0,  -5,   0;
   1,   0,  -1,   0,  -6,   0,   3;
   1,   0,  -2,   0,  -6,   0,   8,   0;
   1,   0,  -3,   0,  -5,   0,  14,   0,  -3;
   1,   0,  -4,   0,  -3,   0,  20,   0, -11,   0;

Cf. A174559.

BB := proc(m,n) if m=0 then if n= 0 then 3 ; else -1; end if; else (3*m-n)*binomial(n+m-1,n)/m ; end if; end proc:
A193002 := proc(n,k) if type(k,'odd') then 0; else (-1)^(1+k/2)*BB(k/2,n-k) ; end if; end proc:
seq(seq(A193002(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 30 2011

Sum_{k=0..n} T(n,k) = A130806(n+5). (row sums)
Sum_{k=0..n} (-1)^(k/2)*T(n,k) = -A000032(n-2). (alternating row sums)
T(n,k) = (-1)^(1+k/2)*BB(k/2,n-k). - R. J. Mathar, Aug 30 2011
T(n,2k) = (-1)^(1+k)*(5-n/k)*binomial(n-k-1,k-1), k > 0. - R. J. Mathar, Aug 30 2011

A237444 Triangle read by rows, T(n,k) is difference of column sum and row sum of natural numbers filled in n x n square.

Original entry on oeis.org

0, 1, -1, 6, 0, -6, 18, 6, -6, -18, 40, 20, 0, -20, -40, 75, 45, 15, -15, -45, -75, 126, 84, 42, 0, -42, -84, -126, 196, 140, 84, 28, -28, -84, -140, -196, 288, 216, 144, 72, 0, -72, -144, -216, -288, 405, 315, 225, 135, 45, -45, -135, -225, -315, -405, 550, 440, 330, 220, 110, 0, -110, -220, -330, -440, -550, 726, 594, 462, 330, 198, 66, -66

Offset: 1

Kival Ngaokrajang, Feb 08 2014

See illustration in links for construction rule.
Column 1 = A002411.
Column 2 = A005564 ,for n >= 3.
Column 3 first differences = A140091.
Nonnegative numbers of this sequence are given by A082375(n,k)*A000217(n), (see example). - Philippe Deléham, Feb 08 2014

			Triangle begins:
n/k   1   2   3   4  5    6   7    8    9   ...
1   0
2   1  -1
3   6   0  -6
4  18   6  -6  18
5  40  20   0 -20 -40
6  75  45  15 -15 -45 -75
7 126  84  42   0 -42 -84 -126
8 196 140  84  28 -28 -84 -140 -196
9 288 216 144  72   0 -72 -144 -216 -288  ...
...
A082375 begins:
0;
1;
2, 0;
3, 1;
4, 2, 0;
5, 3, 1;
6, 4, 2, 0;
7, 5, 3, 1;
8, 6, 4, 2, 0;
9, 7, 5, 3, 1;
.....
A000217 (triangular numbers) begins:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
A082375(n,k)*A000217(n) begins:
0;
1;
6, 0;
18, 6;
40, 20, 0;
75, 45, 15;
126, 84, 42, 0;
196, 140, 84, 28;
288, 216, 144, 72, 0;
405, 315, 225, 135, 45;
... - _Philippe Deléham_, Feb 08 2014

Kival Ngaokrajang, Illustration for n = 1..5

Cf. A000217, A002411, A005564, A114327, A140091.

T(n,k) = - T(n,n-k+1), T(2n+1,n+1)= 0. - Philippe Deléham, Feb 08 2014

T(n+1,k+1) = A114327(n,k)*A000217(n). - Philippe Deléham, Feb 08 2014

cp's OEIS Frontend

A001701 Generalized Stirling numbers.

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A288318 Number T(n,k) of Dyck paths of semilength n such that each positive level has exactly k peaks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A077414 a(n) = n*(n - 1)*(n + 2)/2.

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A055137 Regard triangle of rencontres numbers (see A008290) as infinite matrix, compute inverse, read by rows.

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A093768 Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths.

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A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

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A077414 a(n) = n(n - 1)(n + 2)/2.