A299337 -id:A299337 - cp's OEIS frontend

A299335 Expansion of 1 / ((1 - x)^7*(1 + x)^2).

1, 5, 17, 45, 103, 211, 399, 707, 1190, 1918, 2982, 4494, 6594, 9450, 13266, 18282, 24783, 33099, 43615, 56771, 73073, 93093, 117481, 146965, 182364, 224588, 274652, 333676, 402900, 483684, 577524, 686052, 811053, 954465, 1118397, 1305129, 1517131, 1757063

Offset: 0

Colin Barker, Feb 07 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

Cf. A001769, A060099, A299336, A299337, A299338.

CoefficientList[Series[1/((1 - x)^7 (1 + x)^2), {x, 0, 40}], x] (* or *) LinearRecurrence[{5, -8, 0, 14, -14, 0, 8, -5, 1}, {1, 5, 17, 45, 103, 211, 399, 707, 1190}, 41] (* Robert G. Wilson v, Feb 07 2018 *)

Vec(1 / ((1 - x)^7*(1 + x)^2) + O(x^40))

a(n) = (2*n^6 + 54*n^5 + 575*n^4 + 3060*n^3 + 8468*n^2 + 11376*n + 5760) / 5760 for n even.
a(n) = (2*n^6 + 54*n^5 + 575*n^4 + 3060*n^3 + 8468*n^2 + 11286*n + 5355) / 5760 for n odd.
a(n) = 5*a(n-1) - 8*a(n-2) + 14*a(n-4) - 14*a(n-5) + 8*a(n-7) - 5*a(n-8) + a(n-9) for n>8.

A299336 Expansion of 1 / ((1 - x)^7*(1 + x)^4).

Original entry on oeis.org

1, 3, 10, 22, 49, 91, 168, 280, 462, 714, 1092, 1596, 2310, 3234, 4488, 6072, 8151, 10725, 14014, 18018, 23023, 29029, 36400, 45136, 55692, 68068, 82824, 99960, 120156, 143412, 170544, 201552, 237405, 278103, 324786, 377454, 437437, 504735, 580888, 665896

Offset: 0

Colin Barker, Feb 07 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

Cf. A001769, A060099, A299335, A299337, A299338.

Vec(1 / ((1 - x)^7*(1 + x)^4) + O(x^40))

a(n) = (2*n^6 + 66*n^5 + 860*n^4 + 5640*n^3 + 19568*n^2 + 33984*n + 23040) / 23040 for n even.
a(n) = (2*n^6 + 66*n^5 + 860*n^4 + 5580*n^3 + 18578*n^2 + 28914*n + 15120) / 23040 for n odd.
a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11) for n>10.

A299338 Expansion of 1 / ((1 - x)^7*(1 + x)^6).

Original entry on oeis.org

1, 1, 7, 7, 28, 28, 84, 84, 210, 210, 462, 462, 924, 924, 1716, 1716, 3003, 3003, 5005, 5005, 8008, 8008, 12376, 12376, 18564, 18564, 27132, 27132, 38760, 38760, 54264, 54264, 74613, 74613, 100947, 100947, 134596, 134596, 177100, 177100, 230230, 230230

Offset: 0

Colin Barker, Feb 07 2018

Same as A000579 but with repeated terms.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Cf. A000579, A001769, A060099, A299335, A299336, A299337.

CoefficientList[Series[1/((1-x)^7(1+x)^6),{x,0,50}],x] (* or *) LinearRecurrence[ {1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{1,1,7,7,28,28,84,84,210,210,462,462,924},50] (* Harvey P. Dale, Oct 09 2018 *)

Vec(1 / ((1 - x)^7*(1 + x)^6) + O(x^40))

a(n) = (2*n^6 + 84*n^5 + 1400*n^4 + 11760*n^3 + 51968*n^2 + 112896*n + 92160) / 92160 for n even.
a(n) = (2*n^6 + 72*n^5 + 1010*n^4 + 6960*n^3 + 24278*n^2 + 39048*n + 20790) / 92160 for n odd.
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>12.

A157901 Triangle read by rows: A000012 * A157898.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 3, 6, 10, 8, 8, 3, 9, 16, 24, 16, 16, 4, 12, 28, 40, 56, 32, 32, 4, 16, 40, 80, 96, 128, 64, 64, 5, 20, 60, 120, 216, 224, 288, 128, 128, 5, 25, 80, 200, 336, 560, 512, 640, 256, 256, 6, 30, 110, 280, 616, 896, 1408, 1152, 1408, 512, 512

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 08 2009

Multiplication of the lower triangular matrix A157898 from the left by A000012 means: these are partial column sums of A157898.

			First few rows of the triangle, n>=0:
  1;
  1,  1;
  2,  2,  2;
  2,  4,  4,   4;
  3,  6, 10,   8,   8;
  3,  9, 16,  24,  16,  16;
  4, 12, 28,  40,  56,  32,  32;
  4, 16, 40,  80,  96, 128,  64,  64;
  5, 20, 60, 120, 216, 224, 288, 128, 128;
  5, 25, 80, 200, 336, 560, 512, 640, 256, 256;

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

Cf. A000012, A105635, A157898.
Columns: A004526 (k=0), A002620 (k=1), A006584 (k=2), 4*A096338 (k=3), 8*A177747 (k=4), 16*A299337 (k=5), 32*A178440 (k=6).
Sums include: A105635(n+1) (row), A166486(n+1) (alternating sign diagonal), A232801(n+1) (diagonal).

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
A157071:= func< n,k | (&+[A157898(j+k,k): j in [0..n-k]]) >;
[A157071(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2025

t[n_, k_]:= t[n,k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1,k-1] +2*t[n-1,k] - (2 -Boole[n==2])*t[n-2,k-1]]; (* t = A059576 *)
A157898[n_, k_]:= A157898[n,k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
A157901[n_, k_]:= A157901[n,k]= Sum[A157898[j+k,k], {j,0,n-k}];
Table[A157901[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2025 *)

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n, k): return sum((-1)^(n+k-j)*binomial(n, j+k)*t(j+k, k) for j in range(n-k+1))
def A157071(n,k): return sum(A157898(j+k,k) for j in range(n-k+1))
print(flatten([[A157071(n,k) for k in range(n+1)] for n in range(10)])) # G. C. Greubel, Aug 27 2025

T(n,k) = Sum_{j=0..n} A157898(j,k).

Edited by the Associate Editors of the OEIS, Apr 10 2009

More terms from G. C. Greubel, Aug 27 2025

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A299335 Expansion of 1 / ((1 - x)^7*(1 + x)^2).

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A299336 Expansion of 1 / ((1 - x)^7*(1 + x)^4).

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A299338 Expansion of 1 / ((1 - x)^7*(1 + x)^6).

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A157901 Triangle read by rows: A000012 * A157898.

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