A008288 -id:A008288 - cp's OEIS frontend

A331329 a(n) = binomial(5n, n)hypergeom([-4n, -n], [-5n], -1).

1, 9, 145, 2625, 50049, 982729, 19665841, 398796225, 8166636545, 168502295625, 3497529199185, 72949645000065, 1527671538372225, 32100078290806665, 676451066002195825, 14290577765009652865, 302557549412667613185, 6417968867896642617225, 136371773642235542394385

Offset: 0

Peter Luschny, Jan 31 2020

nonn

Special case of generalized Delannoy numbers (see cross-references):

T(n, k) = binomial(k*n, n)*hypergeom([(1-k)*n, -n], [-k*n], -1).

Seiichi Manyama, Table of n, a(n) for n = 0..747
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Cf. A001850 (k=2), A026000 (k=3), A026001 (k=4), this sequence (k=5), A341491 (k=6).

Cf. A008288, A181675, A341476, A341477.

a[n_] := Binomial[5 n, n] Hypergeometric2F1[-4 n, -n, -5 n, -1];
Array[a, 19, 0]

a(n) ~ sqrt(5 + 21/sqrt(17)) * (349 + 85*sqrt(17))^n / (sqrt(Pi*n) * 2^(5*n + 2)). - Vaclav Kotesovec, Feb 13 2021

A008419 Crystal ball sequence for 9-dimensional cubic lattice.

Original entry on oeis.org

1, 19, 181, 1159, 5641, 22363, 75517, 224143, 598417, 1462563, 3317445, 7059735, 14218905, 27298155, 50250765, 89129247, 152951073, 254831667, 413442773, 654862247, 1014889769, 1541911931, 2300409629, 3375210671, 4876601009, 6946419011, 9765268709

Offset: 0

N. J. A. Sloane

This is row/column 9 of the Delannoy numbers array, A008288, which is the main entry for these numbers, listing many more properties. - Shel Kaphan, Jan 07 2023

T. D. Noe, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
Index entries for crystal ball sequences

Cf. A001847, A001848, A001849.
Cf. A240876.
Row/column 9 of A008288.

CoefficientList[Series[(z + 1)^9/(z - 1)^10, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,19,181,1159,5641,22363,75517,224143,598417,1462563},40] (* Harvey P. Dale, Jul 25 2013 *)

G.f.: (1+x)^9/(1-x)^10.
a(n) = (4*n^9+18*n^8+240*n^7+756*n^6+3612*n^5+7182*n^4+14360*n^3+14724*n^2+ 10134*n+2835)/2835. - Johannes W. Meijer, Jul 14 2013
a(0)=1, a(1)=19, a(2)=181, a(3)=1159, a(4)=5641, a(5)=22363, a(6)=75517, a(7)=224143, a(8)=598417, a(9)=1462563, a(n)=10*a(n-1)-45*a(n-2)+ 120*a(n-3)- 210*a(n-4)+252*a(n-5)-210*a(n-6)+120*a(n-7)-45*a(n-8)+ 10*a(n-9)- a(n-10). - Harvey P. Dale, Jul 25 2013
Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = (1 - 1/2 + 1/3 - ... + 1/9) - log(2). - Peter Bala, Mar 23 2024

A047666 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k) = a(n-1,k-1) + a(n-1,k) + a(n,k-1).

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 10, 10, 4, 5, 17, 25, 17, 5, 6, 26, 52, 52, 26, 6, 7, 37, 95, 129, 95, 37, 7, 8, 50, 158, 276, 276, 158, 50, 8, 9, 65, 245, 529, 681, 529, 245, 65, 9, 10, 82, 360, 932, 1486, 1486, 932, 360, 82, 10, 11, 101, 507, 1537, 2947, 3653

Offset: 1

N. J. A. Sloane

Main diagonal is A002002. Rows give A002522, A047667, A047668, ...

A047666 := proc(n,k) option remember; if n = 1 then k; elif k = 1 then n; else A047666(n-1,k-1)+A047666(n,k-1)+A047666(n-1,k); fi; end;

nmax = 11; a[1, k_] := k; a[n_, 1] := n; a[n_, k_] := a[n, k] = a[n-1, k-1] + a[n, k-1] + a[n-1, k]; Flatten[ Table[ a[n-k+1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Feb 10 2012 *)

T(n, m) = (Sum_{i=1..n-m}(2*i+1)*U(n-i-1, m-1)) + (Sum_{i=1..m} (2*i+1)*U(n-2, m-i)) - U(n-2, m-1) where U(n, m) = A008288(n, m). - Floor van Lamoen, Aug 16 2001

A047671 Square array a(n,k) read by antidiagonals: a(n,1)=1, a(1,k)=1, a(n,k) = 1 + a(n-1,k-1) + a(n-1,k) + a(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 19, 10, 1, 1, 13, 37, 37, 13, 1, 1, 16, 61, 94, 61, 16, 1, 1, 19, 91, 193, 193, 91, 19, 1, 1, 22, 127, 346, 481, 346, 127, 22, 1, 1, 25, 169, 565, 1021, 1021, 565, 169, 25, 1, 1, 28, 217, 862, 1933, 2524, 1933, 862

Offset: 1

N. J. A. Sloane

Main diagonal is A027618. Rows give A003215, A047672, A047673, A047674.

a(n, k) = A008288(n-1, k-1) + A047662(n-1, k-1).

A047671 := proc(n,k) option remember; if n = 1 then 1; elif k = 1 then 1; else 1+A047671(n-1,k-1)+A047671(n,k-1)+A047671(n-1,k); fi; end;

nmax = 12; a[, 1] = 1; a[1, ] = 1; a[n_ /; n > 1, k_ /; k > 1] :=  a[n, k] = 1 + a[n-1, k-1] + a[n-1, k] + a[n, k-1]; Flatten[ Table[ a[n-k , k], {n, 1, nmax}, {k, 1, n-1}]] (* Jean-François Alcover, Jul 19 2012 *)

Description corrected by Henry Bottomley, May 09 2000

A103136 Inverse of the Delannoy triangle.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -6, 10, -5, 1, 22, -38, 22, -7, 1, -90, 158, -98, 38, -9, 1, 394, -698, 450, -194, 58, -11, 1, -1806, 3218, -2126, 978, -334, 82, -13, 1, 8558, -15310, 10286, -4942, 1838, -526, 110, -15, 1, -41586, 74614, -50746, 25150, -9922, 3142, -778, 142, -17, 1, 206098, -370610, 254410, -129050

Offset: 0

Paul Barry, Jan 24 2005

The Delannoy triangle is A008288 viewed as a number triangle. It is then given by the Riordan array (1/(1-x), x(1+x)/(1-x)). The absolute value of A103136 is the Riordan array (1+xS(x),xS(x)) which is the inverse of the signed Delannoy triangle (1/(1+x), x(1-x)/(1+x)).
Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -1, -1, -2, -1, -2, -1, -2, -1, -2, ... ] DELTA [ 1, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938; the unsigned version is given by [ 1, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA [ 1, 0, 0, 0, 0, 0, 0, 0, ... ]. - Philippe Deléham, Jul 08 2005
The unsigned number |T(n,k)| counts Schroeder n-paths whose ascent starting at the initial vertex has length k. A Schroeder n-path is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (upsteps), D=(1,-1) (downsteps) and F=(2,0) (flatsteps) and never going below the x-axis. For example, |T(2,0)| = 2 counts FF, FUD; |T(2,1)| = 3 counts UFD, UDF, UDUD; |T(2,2)| = 1 counts UUDD. - David Callan, Jul 14 2006

			From _Paul Barry_, Apr 29 2009: (Start)
Triangle begins
    1;
   -1,    1;
    2,   -3,    1;
   -6,   10,   -5,    1;
   22,  -38,   22,   -7,    1;
  -90,  158,  -98,   38,   -9,    1;
  394, -698,  450, -194,   58,  -11,    1;
Production matrix is
  -1,  1,
   1, -2,  1,
  -1,  2, -2,  1,
   1, -2,  2, -2,  1,
  -1,  2, -2,  2, -2,  1
The unsigned triangle has production matrix
  1, 1,
  1, 2, 1,
  1, 2, 2, 1,
  1, 2, 2, 2, 1,
  1, 2, 2, 2, 2, 1 (End)

def A103136(dim): # Returns a triangle with 'dim' rows
    M = matrix([[simplify(hypergeometric([-n, n-k], [1], 2))
          for n in range(k+1)] + [0]*(dim-k-1) for k in range(dim)])
    return [row[:n+1] for n, row in enumerate(M.inverse())]
A103136(9)  # Peter Luschny, Nov 16 2023

Riordan array (1-f(x), f(x)) with f(x) = xS(-x), S(x) the g.f. of the large Schroeder numbers A006318. Equivalent to Riordan array (g(x), 1-g(x)) where g(x) = (3+x-sqrt(1+6x+x^2))/2.

G.f.: 1/(1 + (x - xy)/(1 + x/(1 + 2x/(1 + x/(1 + 2x/(1+... (continued fraction). - Paul Barry, Apr 29 2009

A128966 Triangle read by rows of coefficients of polynomials P[n](x) defined by P[0]=0, P[1]=x+1; for n >= 2, P[n]=(x+1)P[n-1]+xP[n-2].

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 8, 20, 20, 8, 1, 1, 10, 34, 50, 34, 10, 1, 1, 12, 52, 104, 104, 52, 12, 1, 1, 14, 74, 190, 258, 190, 74, 14, 1, 1, 16, 100, 316, 552, 552, 316, 100, 16, 1, 1, 18, 130, 490, 1058, 1362, 1058, 490, 130, 18, 1, 1, 20, 164

Offset: 0

N. J. A. Sloane, May 10 2007

A variant of A008288 (they satisfy the same recurrence).

			Triangle begins:
0
1, 1
1, 2, 1
1, 4, 4, 1
1, 6, 10, 6, 1
1, 8, 20, 20, 8, 1
1, 10, 34, 50, 34, 10, 1
1, 12, 52, 104, 104, 52, 12, 1
1, 14, 74, 190, 258, 190, 74, 14, 1
1, 16, 100, 316, 552, 552, 316, 100, 16, 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A163271 (row sums), A110170 (central terms).

Cf. A102413.

a128966 n k = a128966_tabl !! n !! k
a128966_row n = a128966_tabl !! n
a128966_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([0], [1, 1])
-- Reinhard Zumkeller, Jul 20 2013

P[0]:=0;
P[1]:=x+1;
for n from 2 to 14 do
P[n]:=expand((x+1)*P[n-1]+x*P[n-2]);
lprint(P[n]);
lprint(seriestolist(series(P[n],x,200)));
od:

t[n_, k_] := 2^(1-n)*Binomial[n, k]*Sum[Binomial[n, 2*m+1]*HypergeometricPFQ[{-k, -m, k-n}, {1/2-n/2, -n/2}, -1], {m, 0, (n-1)/2}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014, after Max Alekseyev *)

{ T(n,k) = sum(m=0,(n-1)\2, binomial(n,2*m+1) * sum(j=0,m, binomial(m,j) * binomial(n-2*j,k-j) * 2^(2*j+1-n) ) ) } \\ Max Alekseyev, Mar 10 2008

P[n](x) = (x+1) * ( ((x+1+sqrt(x^2+6x+1))/2)^n - ((x+1-sqrt(x^2+6x+1))/2)^n ) / sqrt(x^2+6x+1) - Max Alekseyev, Mar 10 2008

P[n](x) = (x+1) * (sqrt(x)*I)^(n-1) * U[n-1](-I*(x+1)/sqrt(x)/2), where U[n](t) is Chebyshev polynomial of the 2nd kind. - Max Alekseyev, Mar 10 2008

A143411 Square array, read by antidiagonals: form the Euler-Seidel matrix for the sequence {2^kk!} and then divide column k by 2^kk!.

Original entry on oeis.org

1, 3, 1, 13, 5, 1, 79, 33, 7, 1, 633, 277, 61, 9, 1, 6331, 2849, 643, 97, 11, 1, 75973, 34821, 7993, 1225, 141, 13, 1, 1063623, 493825, 114751, 17793, 2071, 193, 15, 1, 17017969, 7977173, 1870837, 292681, 34361, 3229, 253, 17, 1

Offset: 0

Peter Bala, Aug 19 2008

This table is closely connected to the constant 1/sqrt(e). The row, column and diagonal entries of this table occur in series acceleration formulas for 1/sqrt(e). For a similar table based on the differences of the sequence {2^k*k!} and related to the constant sqrt(e), see A143410. For other arrays similarly related to constants see A086764 (for e), A143409 (for 1/e), A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)).

			The Euler-Seidel matrix for the sequence {2^k*k!} begins
  ========================================
  n\k|     0     1     2     3     4     5
  ========================================
  0  |     1     2     8    48   384  3840
  1  |     3    10    56   432  4224
  2  |    13    66   488  4656
  3  |    79   554  5144
  4  |   633  5698
  5  |  6331
  ...
.
  Dividing the k-th column by 2^k*k! gives
  ========================================
  n\k|     0     1     2     3     4     5
  ========================================
  0  |     1     1     1     1     1     1
  1  |     3     5     7     9    11
  2  |    13    33    61    97
  3  |    79   277   643
  4  |   633  2849
  5  |  6331
  ...

G. C. Greubel, Antidiagonals n = 0..50, flattened
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial.

Cf. A008288, A065919 (main diagonal), A076571, A086764, A108625, A143007, A143409, A143410.

A:= func< n,k | (&+[Binomial(n,j)*Factorial(k+j)*2^j/Factorial(k): j in [0..n]]) >; // Array
A143411:= func< n,k | A(n-k,k) >; // antidiagonal triangle
[A143411(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

with combinat: T := (n, k) -> 1/k!*add(2^j*binomial(n,j)*(k+j)!, j = 0..n): for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;

A[n_, k_]:= (1/k!)*Sum[Binomial[n,j]*(k+j)!*2^j, {j,0,n}]; (* array *)
A143411[n_, k_]:= A[n-k,k]; (* antidiagonals *)
Table[A143411[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 05 2023 *)

def A(n,k): return sum(binomial(n,j)*factorial(j+k)*2^j/factorial(k) for j in range(n+1)) # array
def A143411(n,k): return A(n-k,k) # antidiagonal triangle
flatten([[A143411(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

T(n,k) = (1/k!)*Sum_{j = 0..n} 2^j*binomial(n,j)*(k+j)!.
Relation with Poisson-Charlier polynomials c_n(x,a):
T(n,k) = (-1)^n*c_n(-(k+1),1/2).
Recurrence relations:
T(n,k) = 2*n*T(n-1,k) + T(n,k-1);
T(n,k) = 2*(n+k)*T(n-1,k) + T(n-1,k-1);
T(n,k) = 2*(k+1)*T(n-1,k+1) + T(n-1,k).
Recurrence for row n entries: 2*k*T(n,k) = (2*n+2*k-1)*T(n,k-1) + T(n,k-2).
E.g.f. for column k: exp(y)/(1 - 2*y)^(k+1).
E.g.f. for array: exp(y)/(1 - x - 2*y) = (1 + x + x^2 + ...) + (3 + 5*x + 7*x^2 + ...)*y + (13 + 37*x + 61*x^2 + ...)*y^2/2! + ... .
Series acceleration formulas for 1/sqrt(e):
Row n: 1/sqrt(e) = 2^n*n!*(1/T(n,0) - 1/(2*1!*T(n,0)*T(n,1)) + 1/(2^2*2!*T(n,1)*T(n,2)) - 1/(2^3*3!*T(n,2)*T(n,3)) + ...). For example, row 3 gives 1/sqrt(e) = 48*(1/79 - 1/(2*79*277) + 1/(8*277*643) - 1/(48*643*1225) + ...).
Column k: 1/sqrt(e) = (1 - (1/2)/1! + (1/2)^2/2! - ... + (-1/2)^k/k!) + (-1)^(k+1)/(2^k*k!)*( Sum_{n = 0..inf} 2^n*n!/(T(n,k)*T(n+1,k)) ). For example, column 3 gives 1/sqrt(e) = 29/48 + 1/48*( 1/(1*9) + 2/(9*97) + 8/(97*1225) + 48/(1225*17793) + ... ).
Main diagonal: 1/sqrt(e) = 1 - 2*( 1/(1*5) - 1/(5*61) + 1/(61*1225) - ... ). See A065919.

A180668 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 5, 14, 32, 67, 133, 256, 484, 905, 1681, 3110, 5740, 10579, 19481, 35856, 65976, 121377, 223277, 410702, 755432, 1389491, 2555709, 4700720, 8646012, 15902537, 29249369, 53798022, 98950036, 181997539, 334745713

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn13 and Kn23 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

nmax:=31: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+4*n-8 od: seq(a(n),n=0..nmax);

LinearRecurrence[{3,-2,0,-1,1},{0,0,1,5,14},40] (* Harvey P. Dale, Dec 15 2023 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+3)-2 with a(0)=0.
a(n) = sum(A008574(m)*A000073(n-m),m=0..n).
a(n+2) = add(A008288(n-k+2,k+2),k=0..floor(n/2)).
GF(x) = (x^2*(1+x)^2)/((1-x)^2*(1-x-x^2-x^3)).
Contribution from Bruno Berselli, Sep 23 2010: (Start)
a(n) = 2*a(n-1)-a(n-4)+4 for n>4.
a(n)-3*a(n-1)+2a(n-2)+a(n-4)-a(n-5) = 0 for n>4. (End)

A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4n^2-16n+18 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 7, 26, 72, 171, 371, 760, 1500, 2889, 5475, 10266, 19116, 35435, 65495, 120832, 222664, 410017, 754671, 1388650, 2554784, 4699707, 8644907, 15901336, 29248068, 53796617, 98948523, 181995914, 334743972, 615691547

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn14 and Kn24 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-1,2,-1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

Cf. A000073, A005899, A008288.

nmax:=30: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+4*(n-2)^2+2 od: seq(a(n),n=0..nmax);

nxt[{n_,a_,b_,c_}]:={n+1,b,c,a+b+c+4n(n-2)+6}; NestList[nxt,{2,0,0,1},30][[;;,2]] (* or *) LinearRecurrence[{4,-5,2,-1,2,-1},{0,0,1,7,26,72},40] (* Harvey P. Dale, Jul 13 2024 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+4*(n-2)^2+2 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+5)-2-4*n with a(0)=0.
a(n) = Sum_{m=0..n} A005899(m)*A000073(n-m).
a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+3,k+3).
GF(x) = (x^2*(1+x)^3)/((1-x)^3*(1-x-x^2-x^3)).
From Bruno Berselli, Sep 23 2010: (Start)
a(n) = 3*a(n-1)-2a(n-2)-a(n-4)+a(n-5)+8 for n>4.
a(n)-4*a(n-1)+5a(n-2)-2*a(n-3)+a(n-4)-2*a(n-5)+a(n-6) = 0 for n>5. (End)

A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8n^3-48n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 9, 42, 140, 383, 925, 2056, 4316, 8705, 17069, 32810, 62192, 116743, 217673, 404000, 747496, 1380177, 2544865, 4688186, 8631620, 15886111, 29230725, 53776968, 98926372, 181971057, 334716197, 615660634, 1132400520

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn15 and Kn25 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-3,3,-3,1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

nmax:=29: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 od: seq(a(n),n=0..nmax);

RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+(8n^3-48n^2+112n-96)/3},a,{n,30}] (* or *) LinearRecurrence[{5,-9,7,-3,3,-3,1},{0,0,1,9,42,140,383},30] (* Harvey P. Dale, Dec 04 2019 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+7)-(12+4*n+4*n^2) with a(0)=0.
a(n) = sum(A008412(m)*A000073(n-m),m=0..n).
a(n+2) = add(A008288(n-k+4,k+4),k=0..floor(n/2)).
GF(x) = (x^2*(1+x)^4)/((1-x)^4*(1-x-x^2-x^3)).

cp's OEIS Frontend

A331329 a(n) = binomial(5*n, n)*hypergeom([-4*n, -n], [-5*n], -1).

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A008419 Crystal ball sequence for 9-dimensional cubic lattice.

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A047666 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k) = a(n-1,k-1) + a(n-1,k) + a(n,k-1).

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A047671 Square array a(n,k) read by antidiagonals: a(n,1)=1, a(1,k)=1, a(n,k) = 1 + a(n-1,k-1) + a(n-1,k) + a(n,k-1).

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Extensions

A103136 Inverse of the Delannoy triangle.

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A128966 Triangle read by rows of coefficients of polynomials P[n](x) defined by P[0]=0, P[1]=x+1; for n >= 2, P[n]=(x+1)*P[n-1]+x*P[n-2].

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A143411 Square array, read by antidiagonals: form the Euler-Seidel matrix for the sequence {2^k*k!} and then divide column k by 2^k*k!.

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A180668 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.

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A331329 a(n) = binomial(5n, n)hypergeom([-4n, -n], [-5n], -1).

A128966 Triangle read by rows of coefficients of polynomials P[n](x) defined by P[0]=0, P[1]=x+1; for n >= 2, P[n]=(x+1)P[n-1]+xP[n-2].

A143411 Square array, read by antidiagonals: form the Euler-Seidel matrix for the sequence {2^kk!} and then divide column k by 2^kk!.

A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4n^2-16n+18 with a(0)=0, a(1)=0 and a(2)=1.

A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8n^3-48n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.