A034839 -id:A034839 - cp's OEIS frontend

A139459 Triangle read by rows: binomial(3n,3k), 0 <= k <= n.

1, 1, 1, 1, 20, 1, 1, 84, 84, 1, 1, 220, 924, 220, 1, 1, 455, 5005, 5005, 455, 1, 1, 816, 18564, 48620, 18564, 816, 1, 1, 1330, 54264, 293930, 293930, 54264, 1330, 1, 1, 2024, 134596, 1307504, 2704156, 1307504, 134596, 2024, 1, 1, 2925, 296010, 4686825, 17383860, 17383860, 4686825, 296010, 2925, 1

Offset: 0

Gary W. Adamson, Apr 22 2008

ConvOffsStoT transform of the dodecahedral numbers A006566 starting (1, 20, 84, 220,...).
Row sums give A007613.
The matrix inverse starts:
  1;
  -1,1;
  19,-20,1;
  -1513,1596,-84,1;
  315523,-332860,17556,-220,1;
  -136085041,143562965,-7572565,95095,-455,1;
  105261234643,-111045393456,5857368972,-73562060,352716,-816,1; - R. J. Mathar, Mar 22 2013

			First few rows of the triangle are:
  [0] 1;
  [1] 1,   1;
  [2] 1,  20,     1;
  [3] 1,  84,    84,     1;
  [4] 1, 220,   924,   220,     1;
  [5] 1, 455,  5005,  5005,   455,   1;
  [6] 1, 816, 18564, 48620, 18564, 816, 1;
  ...
Row 5 = (1, 220, 924, 220, 1) = ConvOffs transform of (1, 20, 84, 220); where A006566 = (0, 1, 20, 84, 220, 455, ...).

Cf. A006566, A007613, A086645, A034839, A070775, A177808.

Table[Binomial[3*n, 3*k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 01 2025 *)

More terms from Amiram Eldar, Jun 01 2025

A207536 Triangle of coefficients of polynomials u(n,x) jointly generated with A105070; see Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 12, 4, 1, 20, 20, 1, 30, 60, 8, 1, 42, 140, 56, 1, 56, 280, 224, 16, 1, 72, 504, 672, 144, 1, 90, 840, 1680, 720, 32, 1, 110, 1320, 3696, 2640, 352, 1, 132, 1980, 7392, 7920, 2112, 64, 1, 156, 2860, 13728, 20592, 9152, 832, 1, 182, 4004

Offset: 1

Clark Kimberling, Feb 18 2012

Subtriangle of the triangle given by (1, 0, 1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 08 2012

			First seven rows:
  1;
  1,  2;
  1,  6,
  1, 12,   4;
  1, 20,  20,
  1, 30,  60,  8;
  1, 42, 140, 56;
From _Philippe Deléham_, Apr 08 2012: (Start)
(1, 0, 1, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,   0;
  1,  6,   0,  0;
  1, 12,   4,  0, 0;
  1, 20,  20,  0, 0, 0;
  1, 30,  60,  8, 0, 0, 0;
  1, 42, 140, 56, 0, 0, 0, 0; (End)

Cf. A105070.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207536 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A105070 *)

u(n,x) = u(n-1,x) + 2x*v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Apr 08 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x)/(1-2*x+x^2-2*y*x^2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
T(n,k) = A034839(n,k)*2^k = binomial(n,2*k)*2^k . (End)

A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2k,2n-4*k).

Original entry on oeis.org

1, 0, 3, 3, 6, 36, 16, 150, 165, 430, 1071, 1365, 4453, 6258, 14841, 29169, 49941, 115356, 190091, 404811, 750792, 1393956, 2808438, 4988268, 9905746, 18207126, 34231566, 65278964, 119255889, 227648406, 418394087, 782045001, 1457704212, 2681909302

Offset: 0

Seiichi Manyama, Mar 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0,6,6,-15,-18,5,12,-3,32,12,-6,-4,18,-33,26,-15,6,-1).

Cf. A376729, A382300, A382495.

Cf. A034839, A377146, A382230.

[&+[Binomial(k+2, 2)*Binomial(2*k, 2*n-4*k): k in [0..n]]: n in [0..41]]; // Vincenzo Librandi, May 11 2025

Table[Sum[Binomial[k+2,2]*Binomial[2*k, 2*n-4*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)

a(n) = sum(k=0, n\2, binomial(k+2, 2)*binomial(2*k, 2*n-4*k));

my(N=2, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..1} 4^k * binomial(3,2*k) * (1-x^2-x^3)^(3-2*k) * x^(5*k)) / ((1-x^2-x^3)^2 - 4*x^5)^3.

a(n) = 6*a(n-2) + 6*a(n-3) - 15*a(n-4) - 18*a(n-5) + 5*a(n-6) + 12*a(n-7) - 3*a(n-8) + 32*a(n-9) + 12*a(n-10) - 6*a(n-11) - 4*a(n-12) + 18*a(n-13) - 33*a(n-14) + 26*a(n-15) - 15*a(n-16) + 6*a(n-17) - a(n-18).

A382495 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(2k,2n-4*k).

Original entry on oeis.org

1, 0, 4, 4, 10, 60, 30, 300, 335, 1000, 2506, 3500, 11879, 17304, 44220, 88592, 161865, 385704, 660964, 1475100, 2807956, 5459860, 11313094, 20816004, 42774780, 80798128, 157292750, 307887904, 579776799, 1138007940, 2146348214, 4126143900, 7878910238, 14878269368

Offset: 0

Seiichi Manyama, Mar 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0,8,8,-28,-40,28,72,2,16,20,-80,-114,56,-68,0,35,40,-96,128,-110,64,-28,8,-1).

Cf. A376729, A382300, A382494.

Cf. A034839, A382470.

[&+[Binomial(k+3, 3)*Binomial(2*k, 2*n-4*k): k in [0..n]]: n in [0..40]]; // Vincenzo Librandi, May 12 2025

Table[Sum[Binomial[k+3,3]*Binomial[2*k, 2*n-4*k],{k,0,Floor[n/2]}],{n,0,33}] (* Vincenzo Librandi, May 12 2025 *)

a(n) = sum(k=0, n\2, binomial(k+3, 3)*binomial(2*k, 2*n-4*k));

my(N=3, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..2} 4^k * binomial(4,2*k) * (1-x^2-x^3)^(4-2*k) * x^(5*k)) / ((1-x^2-x^3)^2 - 4*x^5)^4.

a(n) = 8*a(n-2) + 8*a(n-3) - 28*a(n-4) - 40*a(n-5) + 28*a(n-6) + 72*a(n-7) + 2*a(n-8) + 16*a(n-9) + 20*a(n-10) - 80*a(n-11) - 114*a(n-12) + 56*a(n-13) - 68*a(n-14) + 35*a(n-16) + 40*a(n-17) - 96*a(n-18) + 128*a(n-19) - 110*a(n-20) + 64*a(n-21) - 28*a(n-22) + 8*a(n-23) - a(n-24).

A384907 Number of permutations of {1..n} with all distinct lengths of maximal anti-runs (not increasing by 1).

Original entry on oeis.org

1, 1, 1, 5, 17, 97, 587, 4291, 33109, 319967, 3106433, 35554459, 419889707, 5632467097, 77342295637, 1201240551077, 18804238105133, 328322081898745, 5832312989183807, 113154541564902427, 2229027473451951265, 47899977701182298255, 1037672943682453127645

Offset: 0

Gus Wiseman, Jun 21 2025

nonn

			The permutation (1,2,4,3,5,7,8,6,9) has maximal anti-runs ((1),(2,4,3,5,7),(8,6,9)), with lengths (1,5,3), so is counted under a(9).
The a(0) = 1 through a(4) = 17 permutations:
  ()  (1)  (2,1)  (1,3,2)  (1,2,4,3)
                  (2,1,3)  (1,3,2,4)
                  (2,3,1)  (1,4,2,3)
                  (3,1,2)  (1,4,3,2)
                  (3,2,1)  (2,1,3,4)
                           (2,1,4,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (2,4,3,1)
                           (3,1,4,2)
                           (3,2,1,4)
                           (3,2,4,1)
                           (3,4,2,1)
                           (4,1,3,2)
                           (4,2,1,3)
                           (4,3,1,2)
                           (4,3,2,1)

Christian Sievers, Table of n, a(n) for n = 0..449

For subsets instead of permutations we have A384177.
For strict partitions we have A384880, for runs A384178.
For partitions we have A384885, for runs A384884.
For runs instead of anti-runs we have A384891.
A010027 counts permutations by maximal anti-runs, for runs A123513.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A072574, A242882, A325325, A328592, A329739, A351202, A356606, A384886, A384892.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]

a(n)=if(n,my(b(n)=sum(i=0,n-1,(-1)^i*(n-i)!*binomial(n-1,i)), d=floor(sqrt(2*n)), p=polcoef(prod(i=1,n,1+x*y^i,1+O(y*y^n)*((1-x^(d+1))/(1-x))),n,y)); sum(i=1,d,b(n+1-i)*i!*polcoef(p,i)),1) \\ Christian Sievers, Jun 22 2025

a(n) = Sum_{k=1..n} ( T(n,k) * A000255(n-k) ) for n>=1, where T(n,k) is the number of compositions of n into k distinct parts (cf. A072574).

a(11) and beyond from Christian Sievers, Jun 22 2025

A385215 Number of maximal sparse submultisets of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 03 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sparse submultisets of the prime indices of n = 8 are {{},{1},{1,1},{1,1,1}}, with maximization {{1,1,1}}. So a(8) = 1.
The sparse submultisets of the prime indices of n = 462 are {{},{1},{2},{4},{5},{1,4},{2,4},{1,5},{2,5}}, with maximization {{1,4},{1,5},{2,4},{2,5}}, so a(462) = 4.
The prime indices of n together their a(n) maximal sparse submultisets for n = 1, 6, 210, 462, 30030, 46410:
  {}  {1,2}  {1,2,3,4}  {1,2,4,5}  {1,2,3,4,5,6}  {1,2,3,4,6,7}
  ------------------------------------------------------------
  {}   {1}     {1,3}      {1,4}       {2,5}          {1,3,6}
       {2}     {1,4}      {1,5}       {1,3,5}        {1,3,7}
               {2,4}      {2,4}       {1,3,6}        {1,4,6}
                          {2,5}       {1,4,6}        {1,4,7}
                                      {2,4,6}        {2,4,6}
                                                     {2,4,7}

This is the maximal case of A166469.
For binary instead of prime indices we have A384883, maximal case of A245564.
The greatest number whose prime indices are one of these submultisets is A385216.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000045, A000071, A001629, A202064, A316476, A319630, A374356, A384890.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
maxq[els_]:=Select[els,Not[Or@@Table[Divisible[oth,#],{oth,DeleteCases[els,#]}]]&];
Table[Length[maxq[Select[Divisors[n],FreeQ[Differences[prix[#]],1]&]]],{n,30}]

a(n) <= A166469(n).

A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 16, 19, 22, 25, 26, 32, 35, 38, 44, 49, 50, 52, 64, 67, 70, 76, 87, 88, 91, 93, 97, 98, 100, 104, 107, 109, 117, 128, 131, 134, 140, 151, 152, 155, 157, 167, 174, 176, 179, 182, 185, 186, 193, 194, 196, 200, 203, 205, 208, 211, 214, 217

Offset: 1

Gus Wiseman, Jul 04 2025

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    8:    1000 ~ {4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   19:   10011 ~ {1,2,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}
   32:  100000 ~ {6}
   35:  100011 ~ {1,2,6}
   38:  100110 ~ {2,3,6}
   44:  101100 ~ {3,4,6}
   49:  110001 ~ {1,5,6}
   50:  110010 ~ {2,5,6}
   52:  110100 ~ {3,5,6}
   64: 1000000 ~ {7}
   67: 1000011 ~ {1,2,7}
   70: 1000110 ~ {2,3,7}
   76: 1001100 ~ {3,4,7}
   87: 1010111 ~ {1,2,3,5,7}
   88: 1011000 ~ {4,5,7}
   91: 1011011 ~ {1,2,4,5,7}
   93: 1011101 ~ {1,3,4,5,7}
   97: 1100001 ~ {1,6,7}
   98: 1100010 ~ {2,6,7}
  100: 1100100 ~ {3,6,7}

The LHS rank statistic is A069010, counted by A034839 (for partitions A384881, A116674).
The RHS rank statistic is A384890, counted by A384893 (for partitions A268193, A384905).
Subsets of this type are counted by A385572, with n A217615.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A000079, A010027, A053538, A210034, A384177, A384879, A384889.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],Length[Split[bpe[#],#2==#1+1&]]==Length[Split[bpe[#],#2!=#1+1&]]&]

is_ok(n)=hammingweight(n)==2*hammingweight(bitand(n,n>>1))+1 \\ Christian Sievers, Jul 18 2025

A034850 Triangular array formed by taking every other term of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 6, 1, 5, 10, 1, 6, 20, 6, 1, 21, 35, 7, 1, 28, 70, 28, 1, 9, 84, 126, 36, 1, 10, 120, 252, 120, 10, 1, 55, 330, 462, 165, 11, 1, 66, 495, 924, 495, 66, 1, 13, 286, 1287, 1716, 715, 78, 1, 14, 364, 2002, 3432, 2002, 364, 14, 1, 105, 1365, 5005, 6435

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  1;
  2;
  1,  3;
  1,  6,  1;
  5, 10,  1;
  6, 20,  6;
  1, 21, 35,  7;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
D. Dumont and J. Zeng, Polynômes d'Euler et les fractions continues de Stieltjes-Rogers, preprint 1996.
D. Dumont and J. Zeng, Polynômes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

Cf. A007318, A034839.

Table[If[k < 0 || k > (Floor[n/4] + Floor[(n + 1)/4]), 0, Binomial[n, 2*k + Mod[Floor[(n + 1)/2], 2]]], {n, 0, 20}, {k, 0, (Floor[n/4] + Floor[(n + 1)/4])}] // Flatten (* G. C. Greubel, Feb 23 2018 *)

{T(n, k) = if( k<0 || k>n\4 + (n+1)\4, 0, binomial(n, 2*k + (n+1)\2%2))}; /* Michael Somos, Feb 11 2004 */

a(n) = A007318(2n) if both are regarded as integer sequences. - Michael Somos, Feb 11 2004

A119462 Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 12, 2, 2, 20, 10, 2, 30, 30, 2, 2, 42, 70, 14, 2, 56, 140, 56, 2, 2, 72, 252, 168, 18, 2, 90, 420, 420, 90, 2, 2, 110, 660, 924, 330, 22, 2, 132, 990, 1848, 990, 132, 2, 2, 156, 1430, 3432, 2574, 572, 26, 2, 182, 2002, 6006, 6006, 2002, 182, 2, 2, 210, 2730, 10010, 12870, 6006, 910, 30

Offset: 0

Emeric Deutsch, May 21 2006

Row n contains 1 + floor(n/2) terms.
Sum of entries in row n is 2^n (A000079).
2*binomial(n-1,2k) is also the number of permutations avoiding both 123 and 132 with k valleys, i.e., positions with w[i]>w[i+1]Lara Pudwell, Dec 19 2018

			T(3,1) = 6 because we have 001, 010, 011, 100, 101 and 110.
Triangle starts:
  1;
  2;
  2,  2;
  2,  6;
  2, 12,  2;
  2, 20, 10;
  2, 30, 30, 2;
  ...

Muniru A Asiru, Rows n=0..150, flattened
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.

Cf. A000079, A002378, A034827, A034839, A057711.

Concatenation([1],Flat(List([1..15],n->List([0..Int(n/2)],k->2*Binomial(n,2*k))))); # Muniru A Asiru, Dec 20 2018

T:=proc(n,k) if n=0 and k=0 then 1 else 2*binomial(n,2*k) fi end: for n from 0 to 15 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

T[0,0]:=1; T[n_,k_]:= 2*Binomial[n,2k]; Table[T[n,k],{n,0,15},{k,0,Floor[n/2]}]//Flatten (* Stefano Spezia, Apr 19 2025 *)

T(n,k) = 2*binomial(n,2k) for n >= 1; T(0,0) = 1.
T(n,k) = 2*T(n-1,k) - T(n-2,k) + T(n-2,k-1) for n >= 3.
G.f.: (1 - z^2 + t*z^2)/(1 - 2*z + z^2 - t*z^2).
T(n,0) = 2 for n >= 1.
T(n,1) = 2*binomial(n,2) = A002378(n-1).
T(n,2) = 2*binomial(n,4) = A034827(n).
T(n,k) = 2*A034839(n-1,k) for n >= 1. [Corrected by Georg Fischer, May 28 2023]
Sum_{k=0..floor(n/2)} k*T(n,k) = A057711(n).

A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} binomial(n+3j,3j)x^j, in row n, column k.

Original entry on oeis.org

1, 1, 2, 1, 7, 1, 1, 16, 10, 1, 30, 45, 5, 1, 50, 141, 50, 1, 1, 77, 357, 266, 28, 1, 112, 784, 1016, 266, 8, 1, 156, 1554, 3139, 1554, 156, 1, 1, 210, 2850, 8350, 6765, 1452, 55, 1, 275, 4917, 19855, 24068, 9042, 880, 11

Offset: 0

Roger L. Bagula, May 30 2010

Every third row is symmetrical.
Row sums are 3^n.
2*k instead of 3*k in the binomial() gives A034839 with alternating rows of A086645.

			1;
1, 2;
1, 7, 1;
1, 16, 10;
1, 30, 45, 5;
1, 50, 141, 50, 1;
1, 77, 357, 266, 28;
1, 112, 784, 1016, 266, 8;
1, 156, 1554, 3139, 1554, 156, 1;
1, 210, 2850, 8350, 6765, 1452, 55;
1, 275, 4917, 19855, 24068, 9042, 880, 11;

Cf. A094531, A111808, A027907.

A178618 := proc(n,k)
    (1-x)^(n+1)*add( binomial(n+3*j,3*j)*x^j,j=0..n+1) ;
    coeftayl(%,x=0,k) ;
end proc:
seq(seq(A178618(n,k),k=0..n),n=0..8) ; # R. J. Mathar, Nov 05 2012

p[x_, n_] = (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[Binomial[n + 3*k, 3*k]*x^k, {k, 0, Infinity}]
Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]

cp's OEIS Frontend

A139459 Triangle read by rows: binomial(3*n,3*k), 0 <= k <= n.

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A207536 Triangle of coefficients of polynomials u(n,x) jointly generated with A105070; see Formula section.

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A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2*k,2*n-4*k).

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A382495 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(2*k,2*n-4*k).

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A384907 Number of permutations of {1..n} with all distinct lengths of maximal anti-runs (not increasing by 1).

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A385215 Number of maximal sparse submultisets of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

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A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

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A034850 Triangular array formed by taking every other term of Pascal's triangle.

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A139459 Triangle read by rows: binomial(3n,3k), 0 <= k <= n.

A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2k,2n-4*k).

A382495 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(2k,2n-4*k).

A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} binomial(n+3j,3j)x^j, in row n, column k.