A349933 -id:A349933 - cp's OEIS frontend

A349936 Central pentanomial coefficients.

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)

A163269 T(n,k) = largest coefficient in the expansion of (1 + ... + x^(n-1))^(2*k).

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jul 24 2009

T(n,k) = number of ways the sums of all components of two 1..n k-vectors can be equal.
T(n,k) is an odd polynomial in n of order 2*k-1.
Examples:
T(n,1) = n.
T(n,2) = (2/3)*n^3 + (1/3)*n.
T(n,3) = (11/20)*n^5 + (1/4)*n^3 + (1/5)*n.
T(n,4) = (151/315)*n^7 + (2/9)*n^5 + (7/45)*n^3 + (1/7)*n.
Table starts:
  1  1    1     1      1 ...
  2  6   20    70    252 ...
  3 19  141  1107   8953 ...
  4 44  580  8092 116304 ...
  5 85 1751 38165 856945 ...
  ...

			For n = 3 and k = 2, (1 + x + x^2)^(2*2) = x^8 + 4*x^7 + 10*x^6 + 16*x^5 + 19*x^4 + 16*x^3 + 10*x^2 + 4*x + 1, whose largest coefficient is T(3,2) = 19.

R. H. Hardin, Table of n, a(n) for n = 1..3240

Removing the leftmost column of A349933 generates this sequence.

Cf. A273975.

T(n,k) = polcoef(sum(i=0, n-1, x^i)^(2*k), k*(n-1)); \\ Michel Marcus, Jan 23 2024

T(n,k) = A273975(2*k, n, (n-1)*k). - Andrey Zabolotskiy, Jan 23 2024

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).

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A349936 Central pentanomial coefficients.

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A163269 T(n,k) = largest coefficient in the expansion of (1 + ... + x^(n-1))^(2*k).

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PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula