A033877 -id:A033877 - cp's OEIS frontend

A080245 Inverse of coordination sequence array A113413.

1, -2, 1, 6, -4, 1, -22, 16, -6, 1, 90, -68, 30, -8, 1, -394, 304, -146, 48, -10, 1, 1806, -1412, 714, -264, 70, -12, 1, -8558, 6752, -3534, 1408, -430, 96, -14, 1, 41586, -33028, 17718, -7432, 2490, -652, 126, -16, 1

Offset: 0

Paul Barry, Feb 13 2003

Formal inverse of A035607 when written as lower triangular matrix 1 2 1 2 4 1 ...

			Rows are {1}, {-2, 1}, {6, -4, 1}, {-22, 16, -6, 1}, ....
From _Paul Barry_, Apr 28 2009: (Start)
Triangle begins
  1,
  -2, 1,
  6, -4, 1,
  -22, 16, -6, 1,
  90, -68, 30, -8, 1,
  -394, 304, -146, 48, -10, 1,
  1806, -1412, 714, -264, 70, -12, 1
Production matrix is
  -2, 1,
  2, -2, 1,
  -2, 2, -2, 1,
  2, -2, 2, -2, 1,
  -2, 2, -2, 2, -2, 1,
  2, -2, 2, -2, 2, -2, 1,
  -2, 2, -2, 2, -2, 2, -2, 1 (End)

Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 7.

Row sums are signed little Schroeder numbers A080243. Diagonal sums are given by A080244.
Cf. A035607, A080243, A080244, A006603, A001003.
Cf. A000007 A084938.
Essentially same triangle as A033877 but with rows read in reversed order.

Essentially the same as the triangle T(n, k), for n>0 and k>0, given by [0, -2, -1, -2, -1, -2, -1, -2, ...] DELTA A000007. Triangle (unsigned) given by [0, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938.

Riordan array ((sqrt(1+6x+x^2)-x-1)/(2x), (sqrt(1+6x+x^2)-x-1)/2).

A378238 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3n+r+k,n)/(3n+r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 14, 0, 1, 6, 32, 134, 0, 1, 8, 54, 324, 1482, 0, 1, 10, 80, 578, 3696, 17818, 0, 1, 12, 110, 904, 6810, 45316, 226214, 0, 1, 14, 144, 1310, 11008, 85278, 583152, 2984206, 0, 1, 16, 182, 1804, 16490, 140936, 1113854, 7769348, 40503890, 0

Offset: 0

Seiichi Manyama, Nov 20 2024

			Square array begins:
  1,      1,      1,       1,       1,       1,       1, ...
  0,      2,      4,       6,       8,      10,      12, ...
  0,     14,     32,      54,      80,     110,     144, ...
  0,    134,    324,     578,     904,    1310,    1804, ...
  0,   1482,   3696,    6810,   11008,   16490,   23472, ...
  0,  17818,  45316,   85278,  140936,  216002,  314700, ...
  0, 226214, 583152, 1113854, 1870352, 2914790, 4320608, ...

Columns k=0..3 give A000007, A144097, A371675, A365843.
T(n,n) gives 1/4 * A370102(n) for n > 0.
Cf. A033877, A071949, A378236, A378237, A378239, A378240.

T(n, k, t=3, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A144097.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+3) for n > 0.

A227505 Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

Original entry on oeis.org

1, 6, 31, 154, 763, 3808, 19197, 97772, 502749, 2607658, 13630635, 71743478, 379949431, 2023314980, 10828048409, 58206726936, 314157742457, 1701817879214, 9249717805207, 50427858276754, 275695956722547, 1511164724634440, 8302888160922965

Offset: 1

Johannes W. Meijer, Jul 15 2013

The terms of this sequence equal the Kn23 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).

Cf. A033877, A006603, A006318, A006319, A227504.

A227505 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if kA227505(n), n = 1..23);
A227505 := proc(n): A006603(n+3) - A006318(n+3) - A006319(n+2) end: A006603 := n ->  add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): A006318 := n -> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): A006319 := proc(n): if n=0 then 1 else A006318(n) - A006318(n-1) fi: end: seq(A227505(n), n=1..23);

a(n) = sum(A033877(n-2*k+2,n-k+3), k=1..floor((n+1)/2)).

a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

A378237 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+3r+k,n)/(n+3r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 10, 0, 1, 6, 24, 74, 0, 1, 8, 42, 188, 642, 0, 1, 10, 64, 350, 1680, 6082, 0, 1, 12, 90, 568, 3234, 16212, 60970, 0, 1, 14, 120, 850, 5440, 31878, 164584, 635818, 0, 1, 16, 154, 1204, 8450, 54888, 328426, 1732172, 6826690, 0, 1, 18, 192, 1638, 12432, 87402, 574848, 3494142, 18728352, 74958914, 0

Offset: 0

Seiichi Manyama, Nov 20 2024

			Square array begins:
   1,     1,      1,      1,      1,      1,       1, ...
   0,     2,      4,      6,      8,     10,      12, ...
   0,    10,     24,     42,     64,     90,     120, ...
   0,    74,    188,    350,    568,    850,    1204, ...
   0,   642,   1680,   3234,   5440,   8450,   12432, ...
   0,  6082,  16212,  31878,  54888,  87402,  131964, ...
   0, 60970, 164584, 328426, 574848, 931770, 1433544, ...

Columns k=0..1 give A000007, A349310.

Cf. A033877, A071949, A378236, A378238, A378239, A378240.

T(n, k, t=1, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A349310.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+3) for n > 0.

A378239 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2n+2r+k,n)/(2n+2r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 12, 0, 1, 6, 28, 100, 0, 1, 8, 48, 248, 968, 0, 1, 10, 72, 452, 2480, 10208, 0, 1, 12, 100, 720, 4680, 26688, 113792, 0, 1, 14, 132, 1060, 7728, 51504, 301648, 1318832, 0, 1, 16, 168, 1480, 11800, 87104, 591312, 3531424, 15732064, 0

Offset: 0

Seiichi Manyama, Nov 20 2024 based on suggestions from Mikhail Kurkov

			Square array begins:
  1,      1,      1,      1,       1,       1,       1, ...
  0,      2,      4,      6,       8,      10,      12, ...
  0,     12,     28,     48,      72,     100,     132, ...
  0,    100,    248,    452,     720,    1060,    1480, ...
  0,    968,   2480,   4680,    7728,   11800,   17088, ...
  0,  10208,  26688,  51504,   87104,  136352,  202560, ...
  0, 113792, 301648, 591312, 1017184, 1621280, 2454256, ...

Columns k=0..4 give A000007, A219534, A371693, A378155, A378156.

Cf. A033877, A071949, A378236, A378237, A378238, A378240.

T(n, k, t=2, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A219534.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-1,k+3) for n > 0.

A006321 Royal paths in a lattice.

Original entry on oeis.org

1, 8, 48, 264, 1408, 7432, 39152, 206600, 1093760, 5813000, 31019568, 166188552, 893763840, 4823997960, 26124870640, 141926904328, 773293020928, 4224773978632, 23139861329456, 127039971696392, 698993630524032, 3853860616119048, 21288789223825648

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

Fourth diagonal of A033877.

1,seq(4*sum(binomial(n,j)*binomial(n+3+j,n-1),j=0..n)/n,n=1..17);

Flatten[{1, RecurrenceTable[{n*(n+4)*a[n] == (5*n^2+14*n+21)*a[n-1] + (5*n^2-4*n+12)*a[n-2] - (n-3)*(n+1)*a[n-3], a[1] == 8, a[2] == 48,a[3] == 264}, a, {n,25}]}] (* Vaclav Kotesovec, Oct 05 2012 *)

a(n) = (4/n)*sum(binomial(n, j)*binomial(n+3+j, n-1), j=0..n) (n>0). - Emeric Deutsch, Aug 19 2004
Recurrence: n*(n+4)*a(n) = (5*n^2+14*n+21)*a(n-1) + (5*n^2-4*n+12)*a(n-2) - (n-3)*(n+1)*a(n-3). - Vaclav Kotesovec, Oct 05 2012
a(n) ~ 2*sqrt(816+577*sqrt(2))*(3+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 05 2012
G.f.: (x^4-8*x^3+16*x^2-8*x+1+sqrt(x^2-6*x+1)*(x-1)*(x^2-4*x+1))/(2*x^4). - Mark van Hoeij, Apr 16 2013

More terms from Vincenzo Librandi, May 03 2013

A144944 Super-Catalan triangle (read by rows) = triangular array associated with little Schroeder numbers (read by rows): T(0,0)=1, T(p,q) = T(p,q-1) if 0 < p = q, T(p,q) = T(p,q-1) + T(p-1,q) + T(p-1,q-1) if -1 < p < q and T(p,q) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 5, 11, 11, 1, 7, 23, 45, 45, 1, 9, 39, 107, 197, 197, 1, 11, 59, 205, 509, 903, 903, 1, 13, 83, 347, 1061, 2473, 4279, 4279, 1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793, 1, 17, 143, 795, 3285, 10717, 28435, 61463, 103049, 103049

Offset: 0

Johannes Fischer (Fischer(AT)informatik.uni-tuebingen.de), Sep 26 2008

			First few rows of the triangle:
  1
  1,  1
  1,  3,  3
  1,  5, 11,  11
  1,  7, 23,  45,  45
  1,  9, 39, 107, 197, 197
  1, 11, 59, 205, 509, 903, 903

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Johannes Fischer and Volker Heun, Finding Range Minima in the Middle: Approximations and Applications, Math. Comput. Sci. 3(1): 17-30 (2010). See Eq. (3.3)
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 8.

Super-Catalan numbers or little Schroeder numbers (cf. A001003) appear on the diagonal.
Generalizes the Catalan triangle (A009766) and hence the ballot Numbers.
Cf. A033877 for a similar triangle derived from the large Schroeder numbers (A006318).
Cf. A010683 (row sums), A186826 (rows reversed).
Cf. A239204, A247623.

a144944 n k = a144944_tabl !! n !! k
a144944_row n = a144944_tabl !! n
a144944_tabl = iterate f [1] where
   f us = vs ++ [last vs] where
     vs = scanl1 (+) $ zipWith (+) us $ [0] ++ us
-- Reinhard Zumkeller, May 11 2013

t[, 0]=1; t[p, p_]:= t[p, p]= t[p, p-1]; t[p_, q_]:= t[p, q]= t[p, q-1] + t[p-1, q] + t[p-1, q-1]; Flatten[Table[ t[p, q], {p,0,6}, {q,0, p}]] (* Jean-François Alcover, Dec 19 2011 *)

@CachedFunction
def t(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (kG. C. Greubel, Mar 11 2023

From G. C. Greubel, Mar 11 2023: (Start)
Sum_{k=0..n} T(n, k) = A010683(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A247623(n). (End)

A227504 Schroeder triangle sums: a(n) = A006603(n+1) - A006318(n+1).

Original entry on oeis.org

1, 4, 17, 74, 335, 1566, 7515, 36836, 183709, 929392, 4758477, 24611950, 128411643, 675051770, 3572165431, 19012868648, 101718917721, 546707554844, 2950563205705, 15983712882930, 86880753686279, 473710078493718, 2590187432233363, 14199709022579788

Offset: 1

Johannes W. Meijer, Jul 15 2013

The terms of this sequence equal the Kn22 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).

Cf. A033877, A006603, A006318, A227505.

A227504 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if kA227504(n), n = 1..24);
A227504 := proc(n): A006603(n+1) - A006318(n+1) end: A006603 := n -> add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2)) / (n-k+2), k= 1.. n/2+1): A006318 := n -> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): seq(A227504(n), n=1..24);

a(n) = sum(A033877(n-2*k+2, n-k+2), k=1..floor((n+1)/2)).

a(n) = A006603(n+1) - A006318(n+1).

A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2r+k,n)/(n+2r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 8, 0, 1, 6, 20, 44, 0, 1, 8, 36, 120, 280, 0, 1, 10, 56, 236, 800, 1936, 0, 1, 12, 80, 400, 1656, 5696, 14128, 0, 1, 14, 108, 620, 2960, 12192, 42416, 107088, 0, 1, 16, 140, 904, 4840, 22592, 92960, 326304, 834912, 0, 1, 18, 176, 1260, 7440, 38352, 176800, 727824, 2572992, 6652608, 0

Offset: 0

Seiichi Manyama, Nov 20 2024

			Square array begins:
   1,     1,     1,     1,      1,      1,      1, ...
   0,     2,     4,     6,      8,     10,     12, ...
   0,     8,    20,    36,     56,     80,    108, ...
   0,    44,   120,   236,    400,    620,    904, ...
   0,   280,   800,  1656,   2960,   4840,   7440, ...
   0,  1936,  5696, 12192,  22592,  38352,  61248, ...
   0, 14128, 42416, 92960, 176800, 308560, 507152, ...

Columns k=0..1 give A000007, A346626.

Cf. A033877, A071949, A378237, A378238, A378239, A378240.

T(n, k, t=1, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A346626.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+2) for n > 0.

A378240 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3n+3r+k,n)/(3n+3r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 18, 0, 1, 6, 40, 234, 0, 1, 8, 66, 540, 3570, 0, 1, 10, 96, 926, 8400, 59586, 0, 1, 12, 130, 1400, 14706, 141876, 1053570, 0, 1, 14, 168, 1970, 22720, 251622, 2528760, 19392490, 0, 1, 16, 210, 2644, 32690, 394152, 4524786, 46815116, 367677090, 0

Offset: 0

Seiichi Manyama, Nov 20 2024

			Square array begins:
  1,       1,       1,       1,       1,        1,        1, ...
  0,       2,       4,       6,       8,       10,       12, ...
  0,      18,      40,      66,      96,      130,      168, ...
  0,     234,     540,     926,    1400,     1970,     2644, ...
  0,    3570,    8400,   14706,   22720,    32690,    44880, ...
  0,   59586,  141876,  251622,  394152,   575402,   801948, ...
  0, 1053570, 2528760, 4524786, 7156128, 10553970, 14867704, ...

Columns k=0..1 give A000007, A364167.

Cf. A033877, A071949, A378236, A378237, A378238, A378239.

T(n, k, t=3, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A364167.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+5). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+5) for n > 0.

cp's OEIS Frontend

A080245 Inverse of coordination sequence array A113413.

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A378238 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*n+r+k,n)/(3*n+r+k) for k > 0.

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A227505 Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

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A378237 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+3*r+k,n)/(n+3*r+k) for k > 0.

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A378239 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2*n+2*r+k,n)/(2*n+2*r+k) for k > 0.

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A006321 Royal paths in a lattice.

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A144944 Super-Catalan triangle (read by rows) = triangular array associated with little Schroeder numbers (read by rows): T(0,0)=1, T(p,q) = T(p,q-1) if 0 < p = q, T(p,q) = T(p,q-1) + T(p-1,q) + T(p-1,q-1) if -1 < p < q and T(p,q) = 0 otherwise.

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A227504 Schroeder triangle sums: a(n) = A006603(n+1) - A006318(n+1).

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A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2*r+k,n)/(n+2*r+k) for k > 0.

A378238 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3n+r+k,n)/(3n+r+k) for k > 0.

A378237 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+3r+k,n)/(n+3r+k) for k > 0.

A378239 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2n+2r+k,n)/(2n+2r+k) for k > 0.

A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2r+k,n)/(n+2r+k) for k > 0.

A378240 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3n+3r+k,n)/(3n+3r+k) for k > 0.