A035343 -id:A035343 - cp's OEIS frontend

A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 19, 4, 1, 1, 70, 141, 44, 5, 1, 1, 252, 1107, 580, 85, 6, 1, 1, 924, 8953, 8092, 1751, 146, 7, 1, 1, 3432, 73789, 116304, 38165, 4332, 231, 8, 1, 1, 12870, 616227, 1703636, 856945, 135954, 9331, 344, 9, 1, 1, 48620, 5196627, 25288120, 19611175, 4395456, 398567, 18152, 489, 10, 1

Offset: 0

Stefano Spezia, Dec 06 2021

			The array begins:
n\s |   0     1     2     3     4
----+----------------------------
  0 |   1     1     1     1     1 ...
  1 |   1     2     3     4     5 ...
  2 |   1     6    19    44    85 ...
  3 |   1    20   141   580  1751 ...
  4 |   1    70  1107  8092 38165 ...
  ...

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 2.

Cf. A000984 (s=1), A082758 (s=2), A005721 (s=3), A349936 (s=4), A063419 (s=5), A270918 (n=s), A163269 (s>0).
Cf. A007318, A008287, A025012, A027907, A035343, A063260.
Cf. A349934, A349935.

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,s_]:=T[2n,s n,s]; Flatten[Table[A[n-s,s],{n,0,9},{s,0,n}]]

A(n, s) = T(2*n, s*n, s), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.

A349936 Central pentanomial coefficients.

Original entry on oeis.org

1, 5, 85, 1751, 38165, 856945, 19611175, 454805755, 10651488789, 251345549849, 5966636799745, 142330448514875, 3408895901222375, 81922110160246231, 1974442362935339179, 47705925773278538281, 1155170746105476171285, 28025439409568101909625, 681077893998769910221225

Offset: 0

Stefano Spezia, Dec 06 2021

Largest coefficient of (Sum_{j=0..4} x^j)^(2*n).

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Central coefficients in triangle A035343.
Column s = 4 in A349933.
Cf. A005721, A063419, A082758.

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; Table[T[2n,4n,4],{n,0,18}]

a(n) = T(2*n, 4*n, 4), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.
a(n) = A035343(2*n, 4*n) = [x^(4*n)] (Sum_{j=0..4} x^j)^(2*n).
From Vaclav Kotesovec, Dec 09 2021: (Start)
Recurrence: 2*n*(2*n - 1)*(3*n - 4)*(4*n - 7)*(4*n - 3)*(4*n - 1)*(6*n - 13)*(6*n - 7)*a(n) = 3*(4*n - 7)*(6*n - 13)*(10584*n^6 - 47628*n^5 + 84190*n^4 - 73965*n^3 + 33531*n^2 - 7272*n + 570)*a(n-1) - 75*(n-1)*(2*n - 3)*(4*n - 5)*(6*n - 1)*(504*n^4 - 2520*n^3 + 4160*n^2 - 2525*n + 476)*a(n-2) + 625*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(3*n - 1)*(4*n - 3)*(6*n - 7)*(6*n - 1)*a(n-3).
a(n) ~ 25^n / sqrt(8*Pi*n). (End)

A064058 Ninth column of quintinomial coefficients.

Original entry on oeis.org

1, 15, 85, 320, 951, 2415, 5475, 11385, 22110, 40612, 71214, 120055, 195650, 309570, 477258, 718998, 1061055, 1537005, 2189275, 3070914, 4247617, 5800025, 7826325, 10445175, 13798980, 18057546, 23422140

Offset: 0

Wolfdieter Lang, Aug 29 2001

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A064057 (eighth column), A000575 (tenth column).

With[{c=8!/4!},Table[(Binomial[n+4,4](n^4+34n^3+451n^2+2874n+1680))/c, {n,0,30}]] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,15,85,320,951,2415,5475,11385,22110},30] (* Harvey P. Dale, Oct 30 2011 *)

a(n) = A035343(n+2, 8) = binomial(n+4, 4)*(n^4+34*n^3+451*n^2+2874*n+1680)/(8!/4!).
G.f.: (1+6*x-14*x^2+11*x^3-3*x^4)/(1-x)^9; numerator polynomial is N5(8, x) from the array A063422.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) with a(0)=1, a(1)=15, a(2)=85, a(3)=320, a(4)=951, a(5)=2415, a(6)=5475, a(7)=11385, a(8)=22110. - Harvey P. Dale, Oct 30 2011
a(n) = C(n+2,2) + 12*C(n+2,3) + 31*C(n+2,4) + 35*C(n+2,5) + 21*C(n+2,6) + 7*C(n+2,7) + C(n+2,8) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A104631 Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

Original entry on oeis.org

0, 1, 4, 18, 80, 365, 1686, 7875, 37080, 175725, 837100, 4004770, 19227924, 92599533, 447118140, 2163837030, 10492874384, 50972030189, 248000853348, 1208335275170, 5894873067200, 28791371852145, 140768761906190

Offset: 0

T. D. Noe, Mar 17 2005

In the triangle of pentanomial coefficients, these numbers are in the column next to the central pentanomial coefficients, A005191. Note that for n>0, n divides a(n). This divisibility property is also true for the triangle of trinomial coefficients, A027907, but apparently for no other triangle of m-nomial coefficients. The quotient a(n)/n is in A104632.

			G.f. = x + 4*x^2 + 18*x^3 + 80*x^4 + 365*x^5 + 1686*x^6 + 7875*x^7 + ... - _Michael Somos_, Aug 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Andrei G. Pronko, Periodic Motzkin chain: Ground states and symmetries, arXiv:2504.00835 [math-ph], 2025. See p. 16.

Cf. A035343 (triangle of pentanomial coefficients).

Cf. A104632.

P:=PolynomialRing(Integers()); [n eq 0 select 0 else Coefficients((1+x+x^2+x^3+x^4)^n)[2*n+2]: n in [0..22]]; // Bruno Berselli, Nov 17 2011

f=1; Table[f=Expand[f(x^4+x^3+x^2+x+1)]; Coefficient[f, x, 2n+1], {n, 30}]

x='x+O('x^30); concat([0], Vec(sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))))) \\ G. C. Greubel, Aug 12 2018

G.f.: sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))). - Mark van Hoeij, Nov 16 2011
From Vaclav Kotesovec, Oct 17 2012: (Start)
Recurrence: 2*(n-1)*(2*n+1)*a(n) = (19*n^2 - 19*n + 2)*a(n-1) + 5*(2*n^2 - 3*n - 1)*a(n-2) - 25*(n-2)*n*a(n-3).
a(n) ~ 5^n/(2*sqrt(Pi*n)). (End)
a(n) = n * A104632(n) for n>=0. - Michael Somos, Aug 12 2018

A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 15, 4, 1, 42, 91, 34, 5, 1, 132, 603, 364, 65, 6, 1, 429, 4213, 4269, 1085, 111, 7, 1, 1430, 30537, 52844, 19845, 2666, 175, 8, 1, 4862, 227475, 679172, 383251, 70146, 5719, 260, 9, 1, 16796, 1730787, 8976188, 7687615, 1949156, 204687, 11096, 369, 10, 1

Offset: 1

Stefano Spezia, Dec 06 2021

			The array begins:
n\s |  1    2     3      4      5
----+----------------------------
  1 |  1    1     1      1      1 ...
  2 |  2    3     4      5      6 ...
  3 |  5   15    34     65    111 ...
  4 | 14   91   364   1085   2666 ...
  5 | 42  603  4269  19845  70146 ...
  ...

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 2.

Cf. A000012 (n=1), A220892 (n=4).
Cf. A000108 (s=1), A099251 (s=2), A264607 (s=3).
Cf. A000027, A007318, A008287, A027907, A035343, A063260.
Cf. A349933.

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,s_]:=T[2n,s n,s]-T[2n,s n+1,s]; Flatten[Table[A[n-s+1,s],{n,10},{s,n}]]

T(n, k, s) = polcoef((sum(i=0, s, x^i))^n, k);
A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s); \\ Michel Marcus, Dec 10 2021

A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.
A(2, n) = A000027(n+1).
A(3, n) = A006003(n+1).

A177960 Numbers of the form A001317(t), excluding those at places of the form t=m*(2^k-1), m>=0, k>=2.

Original entry on oeis.org

3, 5, 17, 51, 257, 1285, 3855, 13107, 65537, 196611, 983055, 1114129, 5570645, 16711935, 50529027, 84215045, 858993459, 4294967297, 21474836485, 219043332147, 365072220245, 1103806595329, 3311419785987

Offset: 1

Vladimir Shevelev, Dec 24 2010

nonn

m-nomial (m>=2) coefficients are coefficients of the polynomial (1+x+...+x^(m-1))^n (n>=0), see A007318 (m=2), A027907 (m=3), A008287 (m=4), A035343 (m=5) etc. For k>=1, consider the triangle of 2^k-nomial coefficients, each entry reduced mod 2, and convert each row of the reduced triangle to a single number by interpreting the sequence of bits as binary representation of a number. This defines sequences A001317 (k=1), A177882 (k=2), A177897 (k=3), etc. The current sequence lists terms of A001317 which are not derived from any of the sequences for k >=2, not from 4-nomial, not from 8-nomial, not from 16-nomial etc.

Conjecture: If for every m>=2, to consider triangle of m-nomial coefficients mod 2 converted to decimal, then the sequence lists terms of A001317 which are not in the union of other sequences for m=3 (A038184), 4 (A177882), 5, 6,...

Cf. A001317, A177882, A177897, A027907, A008287.

Denote by B(n) the number of terms of the sequence among the first n terms of A001317. Then lim_{n->infinity} B(n)/ = Product_{prime p>=2} (1 - 1/(2^p-1)) = A184085.

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A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.

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A349936 Central pentanomial coefficients.

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A064058 Ninth column of quintinomial coefficients.

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A104631 Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

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A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number.

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A177960 Numbers of the form A001317(t), excluding those at places of the form t=m*(2^k-1), m>=0, k>=2.

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