A362125 -id:A362125 - cp's OEIS frontend

A382614 Expansion of 1/(1 - x*(1 + x)^3)^3.

1, 3, 15, 55, 198, 681, 2263, 7341, 23331, 72928, 224814, 684882, 2065346, 6173466, 18310212, 53935350, 157904130, 459755694, 1332010954, 3841812480, 11035346151, 31579747613, 90061069065, 256028590665, 725715896698, 2051465107719, 5784472106577, 16271956316851

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (3,6,-8,-33,-24,39,108,123,84,36,9,1).

Column k=3 of A362125.
Cf. A099234, A382613.
Cf. A000217, A001628, A382406.
Cf. A382615.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^3)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 02 2025

Table[Sum[Binomial[k+2,2]*Binomial[3*k,n-k],{k,0,n}],{n,0,27}] (* Vincenzo Librandi, Apr 02 2025 *)

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

a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(3*k,n-k).
a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3) - 33*a(n-4) - 24*a(n-5) + 39*a(n-6) + 108*a(n-7) + 123*a(n-8) + 84*a(n-9) + 36*a(n-10) + 9*a(n-11) + a(n-12).
G.f.: -1/(x^4+3*x^3+3x^2+x-1)^3. - Vincenzo Librandi, Apr 02 2025

A362126 Expansion of 1/(1 - x*(1+x)^2)^2.

Original entry on oeis.org

1, 2, 7, 18, 47, 118, 290, 702, 1677, 3966, 9300, 21654, 50116, 115388, 264475, 603792, 1373621, 3115222, 7045205, 15892794, 35769390, 80337144, 180091131, 403002108, 900370600, 2008572044, 4474586920, 9955434456, 22123162421, 49107537598, 108891513251

Offset: 0

Seiichi Manyama, Apr 08 2023

Index entries for linear recurrences with constant coefficients, signature (2,3,-2,-6,-4,-1).

Column k=2 of A362125.

Cf. A002478, A362084.

my(N=40, x='x+O('x^N)); Vec(1/(1-x*(1+x)^2)^2)

a(n) = 2*a(n-1) + 3*a(n-2) - 2*a(n-3) - 6*a(n-4) - 4*a(n-5) - a(n-6) for n > 5.

a(n) = Sum_{k=0..n} (-1)^k * binomial(-2,k) * binomial(2*k,n-k) = Sum_{k=0..n} (k+1) * binomial(2*k,n-k).

A381425 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of (1 + x/(1-x)^k)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 5, 1, 0, 1, 4, 12, 10, 1, 0, 1, 5, 22, 37, 18, 1, 0, 1, 6, 35, 92, 102, 30, 1, 0, 1, 7, 51, 185, 345, 258, 47, 1, 0, 1, 8, 70, 326, 880, 1188, 606, 70, 1, 0, 1, 9, 92, 525, 1881, 3851, 3796, 1335, 100, 1, 0, 1, 10, 117, 792, 3563, 10002, 15655, 11364, 2781, 138, 1, 0

Offset: 0

Seiichi Manyama, Apr 07 2025

			Square array begins:
  1, 1,  1,   1,    1,     1,     1, ...
  0, 1,  2,   3,    4,     5,     6, ...
  0, 1,  5,  12,   22,    35,    51, ...
  0, 1, 10,  37,   92,   185,   326, ...
  0, 1, 18, 102,  345,   880,  1881, ...
  0, 1, 30, 258, 1188,  3851, 10002, ...
  0, 1, 47, 606, 3796, 15655, 49468, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000007, A000012, A177787.
Rows n=0..3 give A000012, A001477, A000326, A096000(k-1).
Main diagonal gives A382859.
Cf. A071919, A362125.

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

A(n,k) = Sum_{j=0..k} binomial(k,j) * binomial(n+(k-1)*j-1,n-j).

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A382614 Expansion of 1/(1 - x*(1 + x)^3)^3.

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A362126 Expansion of 1/(1 - x*(1+x)^2)^2.

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A381425 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of (1 + x/(1-x)^k)^k.

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