A055586 -id:A055586 - cp's OEIS frontend

1, 5, 1, 19, 6, 1, 63, 25, 7, 1, 192, 88, 32, 8, 1, 552, 280, 120, 40, 9, 1, 1520, 832, 400, 160, 49, 10, 1, 4048, 2352, 1232, 560, 209, 59, 11, 1, 10496, 6400, 3584, 1792, 769, 268, 70, 12, 1, 26624, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 66304, 43520

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is (((1-z)^3)/(1-2*z)^4)/(1-x*z/(1-z)).
This is the fourth member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
The column sequences appear as A049612(n+1), A055585, A001794, A001789(n+3), A027608, A055586 for m=0..5.

			[0] 1
[1] 5, 1
[2] 19, 6, 1
[3] 63, 25, 7, 1
[4] 192, 88, 32, 8, 1
[5] 552, 280, 120, 40, 9, 1
[6] 1520, 832, 400, 160, 49, 10, 1
[7] 4048, 2352, 1232, 560, 209, 59, 11, 1
Fourth row polynomial (n=3): p(3, x)= 63 + 25*x + 7*x^2 + x^3.

Cf. A007318, A055248, A055249, A055252. Row sums: A049600(n+1, 4).

T := (n, k) -> binomial(n, k)*hypergeom([4, k - n], [k + 1], -1):
for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

a(n, m)=sum(A055252(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055252(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (((1-x)^3)/(1-2*x)^4)*(x/(1-x))^m, m >= 0.

T(n, k) = binomial(n, k)*hypergeom([4, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

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.

Original entry on oeis.org

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

A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.

Original entry on oeis.org

1, 6, 1, 25, 7, 1, 88, 32, 8, 1, 280, 120, 40, 9, 1, 832, 400, 160, 49, 10, 1, 2352, 1232, 560, 209, 59, 11, 1, 6400, 3584, 1792, 769, 268, 70, 12, 1, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1

Offset: 0

Peter Luschny, Sep 23 2024

Triangle T(m,n,k) is a Riordan array of the form ((1-x)^(m-1)*(1-2x)^(-m-1), x/(1-x)), for m = 3. - Igor Victorovich Statsenko, Feb 08 2025

			Triangle starts:
  [0]     1;
  [1]     6,     1;
  [2]    25,     7,     1;
  [3]    88,    32,     8,    1;
  [4]   280,   120,    40,    9,    1;
  [5]   832,   400,   160,   49,   10,    1;
  [6]  2352,  1232,   560,  209,   59,   11,   1;
  [7]  6400,  3584,  1792,  769,  268,   70,  12,  1;
  [8] 16896,  9984,  5376, 2561, 1037,  338,  82, 13,  1;
  [9] 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1;
  ...
Seen as an array of the columns:
  [0] 1,  6, 25,  88,  280,  832,  2352,  6400,  16896, ...
  [1] 1,  7, 32, 120,  400, 1232,  3584,  9984,  26880, ...
  [2] 1,  8, 40, 160,  560, 1792,  5376, 15360,  42240, ...
  [3] 1,  9, 49, 209,  769, 2561,  7937, 23297,  65537, ...
  [4] 1, 10, 59, 268, 1037, 3598, 11535, 34832, 100369, ...
  [5] 1, 11, 70, 338, 1375, 4973, 16508, 51340, 151709, ...
  [6] 1, 12, 82, 420, 1795, 6768, 23276, 74616, 226325, ...

Column k: A055585 (k=0), A001794 (k=1), A001789 (k=2), A027608 (k=3), A055586 (k=4).

Cf. A145018 (diagonal n-2), A375549 (row sums), A049612 (alternating row sums), A122433.

T := (m, n, k) -> binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1);
for n from 0 to 9 do seq(simplify(T(4, n, k)), k = 0..n) od;
# As a binomial sum:
T := (m, n, k) -> add(binomial(m + j, m)*binomial(n + 1, n - (j + k)), j = 0..n-k):
for n from 0 to 9 do [n], seq(T(3, n, k), k = 0..n) od;
# Alternative, generating the array of the columns:
cgf := k -> (1 - x)^(2 - k) / (1 - 2*x)^4:
ser := (k, len) -> series(cgf(k), x, len + 2):
Tcol := (k, len) -> seq(coeff(ser(k, len), x, j), j = 0..len):
seq(lprint([k], Tcol(k, 8)), k = 0..6);

T(m, n, k) = Sum_{j=0..n-k} binomial(m + j, m)*binomial(n + 1, n - (j + k)) for m = 3.

G.f. of column k: (1 - x)^(2 - k) / (1 - 2*x)^4.

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A055584 Triangle of partial row sums (prs) of triangle A055252.

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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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A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.

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