A055580 -id:A055580 - cp's OEIS frontend

A181332 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k nonzero entries in the top row. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

1, 1, 1, 2, 4, 1, 4, 12, 7, 1, 8, 32, 31, 10, 1, 16, 80, 111, 59, 13, 1, 32, 192, 351, 268, 96, 16, 1, 64, 448, 1023, 1037, 530, 142, 19, 1, 128, 1024, 2815, 3598, 2435, 924, 197, 22, 1, 256, 2304, 7423, 11535, 9843, 4923, 1477, 261, 25, 1, 512, 5120, 18943, 34832

Offset: 0

Emeric Deutsch, Oct 13 2010

The sum of entries in row n is A003480(n).
T(n,1) = A001787(n).
T(n,2) = A055580(n-2) (n>=2).
T(n,3) = A055586(n-3) (n>=3).
Sum(k*T(n,k), k>=0) = A054146(n).

			T(2,1)=4 because we have (1/1), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
1,1;
2,4,1;
4,12,7,1;
8,32,31,10,1;
16,80,111,59,13,1;

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Cf. A003480, A001787, A055580, A055586, A181330, A181332.

T := proc (n, k) options operator, arrow: sum(2^j*binomial(k+j, k)*binomial(n-j-2, k-2), j = 0 .. n-k) end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

T(n,k) = sum(2^j*binomial(k+j,k)*binomial(n-2-j,k-2), j=0..n-k).
G.f.: G(t,x) = (1-x)^2/(1-3*x+2*x^2-t*x).
The g.f. of column k is x^k/((1-2*x)^(k+1)*(1-x)^(k-1)) (we have a Riordan array).
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), with T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=2, T(2,1)=4, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham, Nov 26 2013

A367591 Expansion of 1/((1-x) * (1-3*x)^3).

Original entry on oeis.org

1, 10, 64, 334, 1549, 6652, 27064, 105796, 401041, 1483606, 5380840, 19198306, 67559437, 234963352, 808919632, 2760370984, 9346519297, 31429487170, 105039380080, 349114288150, 1154561484781, 3801030845140, 12462203297224, 40705156945324, 132494756301649

Offset: 0

Seiichi Manyama, Nov 24 2023

Index entries for linear recurrences with constant coefficients, signature (10,-36,54,-27).

Partial sums of A027472.

Cf. A055580, A367592.

PARI
```
a(n) = ((2*n^2+4*n+3)*3^(n+1)-1)/8;
```

a(n) = ((2*n^2+4*n+3) * 3^(n+1) - 1)/8.

A196514 Partial sums of A100381.

Original entry on oeis.org

0, 4, 28, 124, 444, 1404, 4092, 11260, 29692, 75772, 188412, 458748, 1097724, 2588668, 6029308, 13893628, 31719420, 71827452, 161480700, 360710140, 801112060, 1769996284, 3892314108, 8522825724, 18589155324, 40399536124, 87509958652

Offset: 0

R. J. Mathar, Oct 03 2011

Like any sequence with a linear recurrence, this has a Pisano period length modulo any k >= 1. The period lengths for this sequence are (modulo k >= 1) 1, 1, 6, 1, 20, 6, 21, 1, 18, 20, 110, 6, 156, 21, 60, 1, 136, 18, 342, 20, ....

Jolley, Summation of Series, Dover (1961), eq (53) page 10.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

[(n^2-n+2)*2^(n+1)-4 : n in [0..30]]; // Vincenzo Librandi, Oct 05 2011

Table[2^n*Binomial[n, 2], {n, 1, 27}] // Accumulate (* Jean-François Alcover, Jun 24 2013 *)
LinearRecurrence[{7,-18,20,-8},{0,4,28,124},30] (* Harvey P. Dale, Jan 12 2016 *)

a(n)=(n^2-n+2)<<(n+1)-4 \\ Charles R Greathouse IV, Oct 05 2011

G.f.: 4*x / ( (x-1)*(2*x-1)^3 ).
a(n) = (n^2 - n + 2)*2^(n+1) - 4 = 4*A055580(n-1).
a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4); a(0)=0, a(1)=4, a(2)=28, a(3)=124. - Harvey P. Dale, Jan 12 2016

A367592 Expansion of 1/((1-x) * (1-4*x)^3).

Original entry on oeis.org

1, 13, 109, 749, 4589, 26093, 140781, 730605, 3679725, 18097645, 87303661, 414459373, 1941186029, 8987616749, 41199871469, 187228759533, 844358755821, 3782116386285, 16838816966125, 74563177424365, 328550363440621, 1441256130749933, 6296699479008749

Offset: 0

Seiichi Manyama, Nov 24 2023

Index entries for linear recurrences with constant coefficients, signature (13,-60,112,-64).

Partial sums of A038845.

Cf. A055580, A367591.

CoefficientList[Series[1/((1 - x)*(1 - 4*x)^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 04 2025 *)

PARI
```
a(n) = ((9*n^2+21*n+14)*4^(n+1)-2)/54;
```

G.f.: 1/((1-x) * (1-4*x)^3).
a(n) = ((9*n^2+21*n+14) * 4^(n+1) - 2)/54.
a(n) = 13*a(n-1) - 60*a(n-2) + 112*a(n-3) - 64*a(n-4). - Wesley Ivan Hurt, Aug 04 2025

cp's OEIS Frontend

A181332 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k nonzero entries in the top row. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

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A367591 Expansion of 1/((1-x) * (1-3*x)^3).

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A196514 Partial sums of A100381.

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A367592 Expansion of 1/((1-x) * (1-4*x)^3).

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