A045721 -id:A045721 - cp's OEIS frontend

A188912 Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).

1, 2, 8, 42, 260, 1816, 13962, 116094, 1029124, 9609144, 93569808, 942642696, 9763181946, 103455616400, 1117379189926, 12264816349938, 136501928050116, 1537591374945704, 17503603786398576, 201128739609458904, 2330480521265639136

Offset: 0

Emanuele Munarini, Apr 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..93

Cf. A005809, A001764, A005809, A006256, A006013, A045721, A188911, A188913

Table[Sum[Binomial[n,k]Binomial[3k,k]/(2k+1)Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}]

makelist(sum(binomial(n,k)*binomial(3*k,k)/(2*k+1)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(3*k, k)*binomial(3*n-3*k, n-k)/((2*k+1)*(2*n-2*k+1)).
E.g.f.: F(1/3,2/3;1,3/2;27*x/4)^2, where F(a1,a2;b1,b2;z) is a hypergeometric series.
From Vaclav Kotesovec, Jun 10 2019: (Start)
Recurrence: 8*n^2*(n+1)*(2*n+1)^2*(9*n^3-54*n^2+84*n-35)*a(n) = 24*n*(324*n^7-2187*n^6+4689*n^5-4185*n^4+1464*n^3+122*n^2-223*n+44)*a(n-1) - 18*(n-1)*(3645*n^7-30618*n^6+96066*n^5-144585*n^4+103662*n^3-21834*n^2-10860*n+4480)*a(n-2) + 2187*(n-2)^2*(n-1)*(3*n-7)*(3*n-5)*(9*n^3-27*n^2+3*n+4)*a(n-3).
a(n) ~ 3^(3*n + 1) / (Pi * n^3 * 2^(n + 1)). (End)

A188913 Binomial convolution of the binomial coefficients bin(3n,n) (A005809) and bin(3n,n)/(2n+1) (A001764).

Original entry on oeis.org

1, 4, 24, 168, 1300, 10896, 97734, 928752, 9262116, 96091440, 1029267888, 11311712352, 126921365298, 1448378629600, 16760687848890, 196237061599008, 2320532776851972, 27676644749022672, 332568471941572944, 4022574792189178080

Offset: 0

Emanuele Munarini, Apr 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..105

Cf. A005809, A001764, A006256, A006013, A045721, A188911, A188912.

Table[Sum[Binomial[n,k]Binomial[3k,k]Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}]

makelist(sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k,0,n),n,0,12);

a(n) = sum(k=0,n,binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1));
vector(66, n, a(n-1)) /* show terms */ /* Joerg Arndt, Apr 13 2011 */

a(n) = sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k=0..n).
E.g.f.: F(1/3,2/3;1/2,1;27*x/4)*F(1/3,2/3;1,3/2;27*x/4), where F(a1,a2;b1,b2;z) is a hypergeometric series.
Recurrence: 8*n^2 * (2*n+1)^2 * (9*n^3 - 54*n^2 + 84*n - 35)*a(n) = 24*(324*n^7 - 2187*n^6 + 4689*n^5 - 4185*n^4 + 1464*n^3 + 122*n^2 - 223*n + 44)*a(n-1) - 18*(3645*n^7 - 30618*n^6 + 96066*n^5 - 144585*n^4 + 103662*n^3 - 21834*n^2 - 10860*n + 4480)*a(n-2) + 2187*(n-2)*(n-1)*(3*n-7)*(3*n-5)*(9*n^3 - 27*n^2 + 3*n + 4)*a(n-3). - Vaclav Kotesovec, Feb 25 2014
a(n) ~ 3^(3*n+1) / (Pi*n^2*2^(n+1)). - Vaclav Kotesovec, Feb 25 2014

A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)binomial(2n - k, n).

Original entry on oeis.org

1, 2, 2, 6, 9, 6, 20, 40, 40, 20, 70, 175, 225, 175, 70, 252, 756, 1176, 1176, 756, 252, 924, 3234, 5880, 7056, 5880, 3234, 924, 3432, 13728, 28512, 39600, 39600, 28512, 13728, 3432, 12870, 57915, 135135, 212355, 245025, 212355, 135135, 57915, 12870

Offset: 0

Peter Luschny, Mar 21 2024

The main diagonal and column 0 of the triangle are the central binomial coefficients, which are the sums of the squares of Pascal's triangle entries. This sum representation can be generalized, and all terms can be seen as sums of coefficients of some polynomials. (See the Example section.)

To see this, consider T(n, k) as the value of the polynomials P(n, k)(x) at x = 1, where P(n, k)(x) = H([-n, -k], [1], x)*H([-n, -n + k], [1], x) and H denotes the hypergeometric sum 2F1. For instance column 0 is given by the row sums of A008459, and column 1 by the row sums of A371401.

			Triangle starts:
[0]    1;
[1]    2,     2;
[2]    6,     9,     6;
[3]   20,    40,    40,    20;
[4]   70,   175,   225,   175,    70;
[5]  252,   756,  1176,  1176,   756,   252;
[6]  924,  3234,  5880,  7056,  5880,  3234,   924;
[7] 3432, 13728, 28512, 39600, 39600, 28512, 13728, 3432;
.
Because of the symmetry, only the sum representation of terms with k <= n/2 are shown.
0:                 [1]
1:               [1+1]
2:             [1+4+1],               [1+4+4]
3:           [1+9+9+1],            [1+9+21+9]
4:      [1+16+36+16+1],       [1+16+66+76+16],        [1+16+76+96+36]
5: [1+25+100+100+25+1], [1+25+160+340+205+25], [1+25+190+460+400+100]

Column 0 and main diagonal are A000984.
Column 1 and subdiagonal are A097070.
Row sums are A045721.
The even bisection of the alternating row sums is A005809.
The central terms are A188662.
Cf. A046899, A092392, A371395, A244038, A371399.
Cf. A008459, A371401.

T := (n, k) -> binomial(k + n, k) * binomial(2*n - k, n):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);

T[n_, k_] := Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, -n +k, 1, 1];
Table[T[n, k], {n, 0, 7}, {k, 0, n}]

T(n, k) = A046899(n, k) * A092392(n, k).
T(n, k) = A046899(n, k) * A046899(n, n - k).
T(n, k) = A092392(n, k) * A092392(n, n - k).
T(n, k) = A371395(n, k) * (n + 1).
T(n, k) = hypergeom([-n, -k], [1], 1) * hypergeom([-n, -n + k], [1], 1).
2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A244038(n).
2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A371399(n).

A371774 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-k+1,n-3k).

Original entry on oeis.org

1, 4, 21, 121, 727, 4473, 27949, 176549, 1124332, 7205511, 46411744, 300183757, 1948255421, 12681654613, 82755728730, 541213820732, 3546268982757, 23276100962571, 153004515241866, 1007131032951572, 6637396253259291, 43791520333601111

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A371773, A371775, A371776.
Cf. A371754, A371770.
Cf. A045721.

a(n) = sum(k=0, n\3, binomial(3*n-k+1, n-3*k));

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(2*n)).
a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [-1-3*n, 1+n, 3/2+n], 27/4). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 2*n*(2*n - 1)*(671*n^4 - 4757*n^3 + 11743*n^2 - 11533*n + 3516)*a(n) = (44957*n^6 - 350256*n^5 + 997889*n^4 - 1236792*n^3 + 563834*n^2 + 39768*n - 60480)*a(n-1) - 10*(19459*n^6 - 156741*n^5 + 461272*n^4 - 575421*n^3 + 211099*n^2 + 106572*n - 60480)*a(n-2) + (93269*n^6 - 753150*n^5 + 2221631*n^4 - 2772678*n^3 + 999800*n^2 + 543408*n - 302400)*a(n-3) - 3*(3*n - 8)*(3*n - 7)*(671*n^4 - 2073*n^3 + 1498*n^2 + 366*n - 360)*a(n-4).
a(n) ~ 3^(3*n + 5/2) / (11 * sqrt(Pi*n) * 2^(2*n)). (End)

A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

Original entry on oeis.org

1, -2, 1, -1, -2, 1, 2, -2, -2, 1, 1, 4, -3, -2, 1, -2, 3, 6, -4, -2, 1, -1, -6, 6, 8, -5, -2, 1, 2, -4, -12, 10, 10, -6, -2, 1, 1, 8, -10, -20, 15, 12, -7, -2, 1, -2, 5, 20, -20, -30, 21, 14, -8, -2, 1, -1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 03 2006

			Triangular sequence begins as:
   1;
  -2,   1;
  -1,  -2,   1;
   2,  -2,  -2,   1;
   1,   4,  -3,  -2,   1;
  -2,   3,   6,  -4,  -2,   1;
  -1,  -6,   6,   8,  -5,  -2,  1;
   2,  -4, -12,  10,  10,  -6, -2,  1;
   1,   8, -10, -20,  15,  12, -7, -2,  1;
  -2,   5,  20, -20, -30,  21, 14, -8, -2,  1;
  -1, -10,  15,  40, -35, -42, 28, 16, -9, -2, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A005809, A045721, A066170.
Row reversal of: A374439.
Columns are related to: A000034 (k=0), A029578 (k=1), A131259 (k=2).
Diagonals are related to: A113679 (k=n-1), A022958 (k=n-2), A005843 (k=n-3), A000217 (k=n-4), -A002378 (k=n-5).
Sums include: (-1)^floor((n+1)/2)*A016116 (signed diagonal), A057079 (row), A119910 (signed row), (-1)^n*A130706 (diagonal).

function T(n,k) // T = A124038
  if k lt 0 or k gt n then return 0;
  elif k ge n-2 then return k-n + (-1)^(n+k);
  else return T(n-1,k-1) - T(n-2,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 22 2025

(* First program *)
t[n_, m_, d_]:= If[n==m && n>1 && m>1, x, If[n==m-1 || n==m+1, -1, If[n==m== 1, x-2, 0]]];
M[d_]:= Table[t[n,m,d], {n,d}, {m,d}];
Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d,10}][[d]], x], {d,10}]]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k>n-2, k-n+(-1)^(n-k), T[n-1, k- 1] -T[n-2,k]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 22 2025 *)

@CachedFunction
def A124038(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A124038(n-1,k) if n==1 else 0
    return A124038(n-1,k-1) - A124038(n-2,k) - h
for n in (0..9): [A124038(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

from sage.combinat.q_analogues import q_stirling_number2
def A124038(n,k): return (1 + ((n-k)%2))*q_stirling_number2(n+1, n-k+1, -1)
print(flatten([[A124038(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 22 2025

From G. C. Greubel, Jan 22 2025: (Start)
T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
T(n, k) = (-1)^floor((n-k+1)/2)*(1 + (n-k mod 2))*qStirling2(n+1, n-k+1,-1).
T(2*n, n) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)* A045721((n-1)/2) ). (End)

Edited by G. C. Greubel, Jan 22 2025

A252355 a(n) = sum_{k = 0..n-1} (-1)^kC(2n-1,k)*C(n-1,k), n>0.

Original entry on oeis.org

1, -2, 1, 8, -29, 34, 92, -512, 919, 818, -9151, 22472, -2924, -156872, 513736, -443392, -2457281, 11094658, -16502221, -31859752, 226433243, -475853006, -217535264, 4333621888, -12126499804, 5346234424

Offset: 1

L. Edson Jeffery, Dec 17 2014

For each n > 0, a(n) is an integer such that A234839(p-n) == 2^(2 - 3*n)*a(n) (mod p), for all primes p >= 2*n+1 [Chamberland, et al., Thm. 2.3].

Marc Chamberland and Karl Dilcher, A Binomial Sum Related to Wolstenholme's Theorem, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672.

Cf. A000040 (primes), A234839, A045721.

a[n_] := Sum[(-1)^k*Binomial[2*n - 1, k]*Binomial[n - 1, k], {k, 0, n - 1}]; Table[a[n], {n, 26}]

a(n) = sum(k=0, n, (-1)^k*binomial(2*n-1,k)*binomial(n-1,k)); \\ Michel Marcus, Jan 13 2016

a(n) = _2F_1(1-2*n,1-n;1;-1), n>0.
Recurrence: 2*(n-1)*(2*n-1)*(7*n-11)*a(n) = -(91*n^3 - 325*n^2 + 368*n - 128)*a(n-1) - 16*(n-2)*(2*n-3)*(7*n-4)*a(n-2). - Vaclav Kotesovec, Dec 17 2014
Lim sup n->infinity |a(n)|^(1/n) = 2*sqrt(2). - Vaclav Kotesovec, Dec 17 2014
exp( Sum_{n >= 1} 2*a(n)*x^n/n ) = 1 + 2*x - 2*x^3 + 4*x^4 - 2*x^5 - 12*x^6 + 40*x^7 - 44*x^8 - 98*x^9 + 520*x^10 - 882*x^11 - 640*x^12 + ... appears to have integer coefficients. - Peter Bala, Jan 04 2016

A387091 a(n) = binomial(9*n+1,n).

Original entry on oeis.org

1, 10, 171, 3276, 66045, 1370754, 28989675, 621216192, 13442126049, 293052087900, 6426898010533, 141629804643600, 3133614810784185, 69566517009302868, 1548833316392624625, 34569147570568156800, 773240476721553042345, 17328840976366636057110

Offset: 0

Seiichi Manyama, Aug 16 2025

Paolo Xausa, Table of n, a(n) for n = 0..700

Cf. A001700, A045721, A052203, A079589, A079590.

Cf. A062994, A169958.

A387091[n_] := Binomial[9*n + 1, n]; Array[A387091, 20, 0] (* Paolo Xausa, Aug 20 2025 *)

PARI
```
a(n) = binomial(9*n+1, n);
```

a(n) = Sum_{k=0..n} binomial(9*n-k,n-k).
G.f.: 1/(1 - x*g^7*(9+g)) where g = 1+x*g^9 is the g.f. of A062994.
G.f.: g^2/(9-8*g) where g = 1+x*g^9 is the g.f. of A062994.
G.f.: B(x)^2/(1 + 8*(B(x)-1)/9), where B(x) is the g.f. of A169958.
D-finite with recurrence +128*n*(8*n-5)*(4*n-1)*(8*n+1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n) -81*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n+1)*(3*n-2)*(9*n-4)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Aug 19 2025
a(n) ~ 3^(18*n+3) / (sqrt(Pi*n) * 2^(24*n+5)). - Vaclav Kotesovec, Aug 20 2025

A096793 Triangle read by rows: a(n,k) is the number of Dyck n-paths containing k odd-length ascents.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 3, 0, 10, 0, 1, 0, 21, 0, 20, 0, 1, 12, 0, 84, 0, 35, 0, 1, 0, 120, 0, 252, 0, 56, 0, 1, 55, 0, 660, 0, 630, 0, 84, 0, 1, 0, 715, 0, 2640, 0, 1386, 0, 120, 0, 1, 273, 0, 5005, 0, 8580, 0, 2772, 0, 165, 0, 1, 0, 4368, 0, 25025, 0, 24024, 0, 5148, 0, 220, 0, 1

Offset: 0

David Callan, Aug 17 2004

a(n,k)=0 unless k and n have the same parity and 0 <= k <= n.
From Emeric Deutsch, Oct 05 2008: (Start)
Sum_{k=0..n} k*a(n,k) = A014300(n).
For the case of even-length ascents see A143950. (End)

			Table begins
.
n |k = 0    1    2    3    4    5    6    7    8
--+---------------------------------------------
0 |    1
1 |    0,   1
2 |    1,   0,   1
3 |    0,   4,   0,   1
4 |    3,   0,  10,   0,   1
5 |    0,  21,   0,  20,   0,   1
6 |   12,   0,  84,   0,  35,   0,   1
7 |    0, 120,   0, 252,   0,  56,   0,   1
8 |   55,   0, 660,   0, 630,   0,  84,   0,   1
.
a(4,0)=3 because the Dyck 4-paths containing no odd-length ascents are UUUUDDDD,UUDUUDDD,UUDDUUDD.

The nonzero entries in column k=0 give A001764, in k=1 give A045721, in k=2 give A090763. The row sums are the Catalan numbers A000108.

Cf. A143950. - Emeric Deutsch, Oct 05 2008

bi[n_, k_] := If[IntegerQ[k], Binomial[n, k], 0]; TableForm[Table[bi[(n+k)/2, (n-k)/2]bi[(3n-k)/2+1, (n+k)/2]/((3n-k)/2+1), {n, 0, 10}, {k, 0, n}]]

a(n, k) = binomial((n+k)/2, (n-k)/2)*binomial((3n-k)/2+1, (n+k)/2)/((3n-k)/2+1).
Equivalently, a(2n+k, k) = binomial(3n+k, k)*T(n) where T(n) = binomial(3n, n)/(2n+1) is A001764. Proof: Given a Dyck (2n+k)-path with k ascents of odd length, delete the peaks (UD) that terminate odd-length ascents. This is a mapping to Dyck (2n)-paths all of whose ascents have even length; there are T[n] such paths. The mapping is clearly onto and is binomial(3n+k, k)-to-1 as follows. A Dyck (2n)-path all of whose ascents have even length has exactly 3n+1 vertices that are (i) not incident with an upstep, or (ii) incident with an upstep and at even distance (possibly 0) from the start of the ascent they lie in. The k deleted UDs can be inserted arbitrarily at these vertices, repetition allowed, to get the preimages -- binomial(3n+k, k) choices.
G.f.: G(z, t) + H(z, t) where G satisfies G^3*(t^2 - 1)*z^2 - G^2*t*z*(2 + t*z) + G*(1 + 2*t*z) - 1 = 0 and H satisfies H^3*(t^2 - 1)*z^2 + H^2*t*z*(2 + t*z) - H*t^2*(1 - t*z) + t^3*z = 0. Here z marks size (n) and t marks number of odd-length ascents (k). G is gf for paths that start with an even-length ascent and H is gf for paths that start with an odd-length ascent. - David Callan, Sep 03 2005
From Emeric Deutsch, Oct 05 2008: (Start)
G.f. G=G(t,z) satisfies G = 1 + zG(t + zG)/(1 - z^2*G^2).
The trivariate g.f. H=H(t,s,z), where t(s) marks odd-length (even-length) ascents satisfies H = 1 + zH(t+szH)/(1-z^2*H^2). (End)

A137207 Number of exceptional sets of roots of type D_n. Also the number of unordered factorizations of the Coxeter element.

Original entry on oeis.org

12, 87, 584, 3835, 25008, 162792, 1060048, 6910695, 45119100, 295038315, 1932260256, 12673336052, 83236707232, 547388545740, 3604063891104, 23755630474079, 156740823815940, 1035157282013085, 6842413166034600, 45265133475699795, 299671339559444160, 1985322768625822080

Offset: 3

F. Chapoton, Mar 05 2008

nonn

			a(3)=12 because D3 is the same as A3.

F. Chapoton Cluster algebras

Cf. A001764 for type A, A045721 for type B.

modu_NC_D:=proc(n) begin (16*n*n-41*n+24)/n/(2*n-1)*binomial(3*n-5,n-2) end;

def A137207(n):
    return (16*n*n-41*n+24)*binomial(3*n-5,n-2)/n/(2*n-1)

a(n) = (2*(n-1)/(2*n-1))*binomial(3*n-3,n-1)-binomial(3*n-5,n-2)+4*binomial(3*n-3,n-3).

a(n) = (16*n^2-41*n+24)/(n*(2*n-1))*binomial(3*n-5,n-2).

a(22)-a(24) from Stefano Spezia, Feb 29 2024

A252501 Triangle T read by rows: T(n,k) = binomial(2n+1,k)binomial(n,k), n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 3, 1, 10, 10, 1, 21, 63, 35, 1, 36, 216, 336, 126, 1, 55, 550, 1650, 1650, 462, 1, 78, 1170, 5720, 10725, 7722, 1716, 1, 105, 2205, 15925, 47775, 63063, 35035, 6435, 1, 136, 3808, 38080, 166600, 346528, 346528, 155584, 24310

Offset: 0

L. Edson Jeffery, Dec 17 2014

			Triangle T begins:
.1
.1.....3
.1....10.....10
.1....21.....63......35
.1....36....216.....336......126
.1....55....550....1650.....1650......462
.1....78...1170....5720....10725.....7722.....1716
.1...105...2205...15925....47775....63063....35035.....6435
.1...136...3808...38080...166600...346528...346528...155584...24310

Marc Chamberland and Karl Dilcher, A Binomial Sum Related to Wolstenholme's Theorem, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672 (see Thm. 2.3).

Cf. A000012 (col. 1), A014105 (col. 2), A001700 (diag), A045721 (row sums).

Cf. A234839, A252355.

Flatten[Table[Binomial[2*n + 1, k]*Binomial[n, k], {n, 0, 8}, {k, 0, n}]] (* Replace Flatten[] with Grid[] to get the triangle. *)

cp's OEIS Frontend

A188912 Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).

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A188913 Binomial convolution of the binomial coefficients bin(3n,n) (A005809) and bin(3n,n)/(2n+1) (A001764).

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A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)*binomial(2*n - k, n).

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A371774 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-k+1,n-3*k).

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A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

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A252355 a(n) = sum_{k = 0..n-1} (-1)^k*C(2*n-1,k)*C(n-1,k), n>0.

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A387091 a(n) = binomial(9*n+1,n).

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A096793 Triangle read by rows: a(n,k) is the number of Dyck n-paths containing k odd-length ascents.

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A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)binomial(2n - k, n).

A371774 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-k+1,n-3k).

A252355 a(n) = sum_{k = 0..n-1} (-1)^kC(2n-1,k)*C(n-1,k), n>0.

A252501 Triangle T read by rows: T(n,k) = binomial(2n+1,k)binomial(n,k), n>=0, 0<=k<=n.