A055581 -id:A055581 - cp's OEIS frontend

1, 4, 1, 13, 5, 1, 38, 18, 6, 1, 104, 56, 24, 7, 1, 272, 160, 80, 31, 8, 1, 688, 432, 240, 111, 39, 9, 1, 1696, 1120, 672, 351, 150, 48, 10, 1, 4096, 2816, 1792, 1023, 501, 198, 58, 11, 1, 9728, 6912, 4608, 2815, 1524, 699, 256, 69, 12, 1, 22784, 16640, 11520

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)^2)/(1-2*z)^3)/(1-x*z/(1-z)).
This is the third 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 A049611(n+1), A001793, A001788, A055580, A055581, A055582, A055583 for m=0..6.

			[0] 1
[1] 4, 1
[2] 13, 5, 1
[3] 38, 18, 6, 1
[4] 104, 56, 24, 7, 1
[5] 272, 160, 80, 31, 8, 1
[6] 688, 432, 240, 111, 39, 9, 1
[7] 1696, 1120, 672, 351, 150, 48, 10, 1
Fourth row polynomial (n = 3): p(3, x) = 38 + 18*x + 6*x^2 + x^3.

Cf. A007318, A055248, A055249. Row sums: A049612(n+1)= A055584(n, 0).

T := (n, k) -> binomial(n, k)*hypergeom([3, 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(A055249(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)+ A055249(n, m), n >= m >= 0, a(n, m) := 0 if n

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

T(n, k) = binomial(n, k)*hypergeom([3, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

A055582 Sixth column of triangle A055252.

Original entry on oeis.org

1, 9, 48, 198, 699, 2223, 6562, 18324, 49029, 126837, 319332, 786258, 1900351, 4521771, 10616598, 24641280, 56622825, 128974545, 291503800, 654311070, 1459617411, 3238002279, 7147093578, 15703473708, 34359737869, 74893491693

Offset: 0

Wolfdieter Lang, May 26 2000

Michael De Vlieger, Table of n, a(n) for n = 0..3296
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (9, -33, 63, -66, 36, -8).

Cf. A055252, A055249, A034009, partial sums of A055581.

CoefficientList[Series[1/(((1 - 2 x)^3) (1 - x)^3), {x, 0, 25}], x] (* Michael De Vlieger, Apr 23 2020 *)

G.f.: 1/(((1-2*x)^3)*(1-x)^3).
a(n)= A055252(n+5, 5). a(n)= sum(a(j), j=0..n-1)+A034009(n), n >= 1.
a(n)= (n^2 - 3*n + 8)*(2^(n+3) -1)/2 - 9*(n+3). [Yahia Kahloune, Aug 11 2013]

A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 6, 3, 4, 3, 1, 0, 6, 6, 6, 4, 1, 10, 6, 9, 10, 9, 5, 1, 0, 10, 12, 15, 16, 13, 6, 1, 15, 10, 16, 21, 25, 25, 18, 7, 1, 0, 15, 20, 28, 36, 41, 38, 24, 8, 1, 21, 15, 25, 36, 49, 61, 66, 56, 31, 9, 1, 0, 21, 30, 45, 64, 85, 102, 104, 80, 39, 10, 1, 28, 21, 36, 55, 81, 113, 146, 168, 160, 111, 48, 11, 1

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(2n, 0) = T(n, 3) with T(2n, 0) = T(n, m) for some other value of m would change the generating function to the coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^m. This would produce A058393, A058394, A057884 (and effectively A007318).

			The array T(n, k) starts:
[0] 1, 0,  3,   0,   6,   0,  10,    0,   15,    0, ...
[1] 1, 1,  3,   3,   6,   6,  10,   10,   15,   15, ...
[2] 1, 2,  4,   6,   9,  12,  16,   20,   25,   30, ...
[3] 1, 3,  6,  10,  15,  21,  28,   36,   45,   55, ...
[4] 1, 4,  9,  16,  25,  36,  49,   64,   81,  100, ...
[5] 1, 5, 13,  25,  41,  61,  85,  113,  145,  181, ...
[6] 1, 6, 18,  38,  66, 102, 146,  198,  258,  326, ...
[7] 1, 7, 24,  56, 104, 168, 248,  344,  456,  584, ...
[8] 1, 8, 31,  80, 160, 272, 416,  592,  800, 1040, ...
[9] 1, 9, 39, 111, 240, 432, 688, 1008, 1392, 1840, ...

Rows are A000217 with zeros, A008805, A002620, A000217, A000290, A001844, A005899.
Columns are A000012, A001477, A016028.
Diagonals include A058396, A049611, A001793, A001788, A055580, A055581, A055582.
The triangle A055252 also appears in half of the array.

gf := n -> (1 + x)^n / (1 - x^2)^3: ser := n -> series(gf(n), x, 20):
seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..9)), n = 0..9); # Peter Luschny, Apr 12 2023

T[0, k_] := If[OddQ[k], 0, (k+2)(k+4)/8];
T[n_, k_] := T[n, k] = If[k == 0, 1, T[n-1, k-1] + T[n-1, k]];
Table[T[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 13 2023 *)

T(n, k) = T(n-1, k-1) + T(n, k-1) with T(0, k) = 1, T(2*n, 0) = T(n, 3) and T(2*n + 1, 0) = 0. Coefficient of x^n in expansion of (1 + x)^k / (1 - x^2)^3.

A119808 Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).

Original entry on oeis.org

1, 2, 1, 4, 5, 8, 17, 2, 16, 49, 16, 32, 129, 78, 4, 64, 321, 300, 44, 128, 769, 1002, 280, 8, 256, 1793, 3048, 1352, 112, 512, 4097, 8678, 5500, 880, 16, 1024, 9217, 23524, 19892, 5120, 272, 2048, 20481, 61410, 66032, 24600, 2544, 32, 4096, 45057, 155616, 205360

Offset: 0

Emeric Deutsch, May 25 2006

Row n contains 1+ceiling(n/2) terms. T(n,0) = 2^n (A000079). T(n,1) = 1+(n-1)*2^n = A000337(n) for n>=1. T(n,2) = 2*A055581(n-3) (n>=3). Sum(k*T(n,k),k>=0) = A081038(n-1).

			T(2,1) = 5 because we have 00, 01, 02, 10 and 20.
Triangle starts:
1;
2,1;
4,5;
8,17,2;
16,49,16;
32,129,78,4;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000079, A000337, A055581, A081038, A034839.

G:=(1-z+t*z)/(1-3*z+2*z^2-2*t*z^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..ceil(n/2)) od; # yields sequence in triangular form

nn=15;f[list_]:=Select[list,#>0&]; a = y x/(1-x) +1;Map[f,CoefficientList[ Series[a/(1-2x a),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 19 2012 *)

G.f.: (1-z+tz)/(1-3z+2z^2-2tz^2). G.f. of column k: 2^(k-1)*z^(2k-1)*/ [(1-z)^k*(1-2z)^(k+1)] (k>=1). Recurrence relation: T(n,k) = 3T(n-1,k) -2T(n-2,k) +2T(n-2,k-1) for n>=2.

A121466 Triangle read by rows: T(n,k) = is the number of directed column-convex polyominoes of area n having along the lower contour exactly k reentrant corners, i.e., a vertical step that is followed by a horizontal step (n >= 1, k >= 0).

Original entry on oeis.org

1, 2, 4, 1, 8, 5, 16, 17, 1, 32, 49, 8, 64, 129, 39, 1, 128, 321, 150, 11, 256, 769, 501, 70, 1, 512, 1793, 1524, 338, 14, 1024, 4097, 4339, 1375, 110, 1, 2048, 9217, 11762, 4973, 640, 17, 4096, 20481, 30705, 16508, 3075, 159, 1, 8192, 45057, 77808, 51340, 12918

Offset: 1

Emeric Deutsch, Aug 02 2006

Also number of nondecreasing Dyck paths of semilength n and such that there are k positive differences in the sequence of the valley altitudes, preceded by a 0. Example: T(5,2)=1 because we have UUDUUDUDDD, where U=(1,1) and D=(1,-1) (the valleys are at the altitudes 1 and 2 with two "jumps" in the sequence 0,1,2).
Row n has ceiling(n/2) terms.
Row sums are the odd-subscripted Fibonacci numbers (A001519).

			T(5,2)=1 because we have the directed column-convex polyomino [(0,2),(1,3),(2,3)] (here the j-th pair gives the lower and upper levels of the j-th column).
Triangle starts:
   1;
   2;
   4,   1;
   8,   5;
  16,  17,   1;
  32,  49,   8;
  64, 129,  39,   1;

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 11, 19.

Cf. A000079, A000337, A001519, A001870, A055581.

with(combinat): T:=(n,k)->add(2^j*binomial(n-k-2-j,k-1)*binomial(k+j,k),j=0..n-2*k-1): for n from 0 to 15 do seq(T(n,k),k=0..ceil(n/2)-1) od; # yields sequence in triangular form

T(n,0) = 2^(n-1) = A000079(n-1).
T(n,1) = 1 + (n-3)*2^(n-2) = A000337(n-2).
T(n,2) = A055581(n-5).
Sum_{k=0..ceiling(n/2)-1} k*T(n,k) = A001870(n-3).
T(n,k) = Sum_{j=0..n-2*k-1} 2^j*binomial(n-k-2-j,k-1)*binomial(k+j,k) for k >= 1; T(n,0) = 2^(n-1).
G.f.: G(t,z) = z(1-z)/(1-3z+2z^2-tz^2).

Showing 1-5 of 5 results.

cp's OEIS Frontend

A055252 Triangle of partial row sums (prs) of triangle A055249.

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A055582 Sixth column of triangle A055252.

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A058395 Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.

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A119808 Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).

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A121466 Triangle read by rows: T(n,k) = is the number of directed column-convex polyominoes of area n having along the lower contour exactly k reentrant corners, i.e., a vertical step that is followed by a horizontal step (n >= 1, k >= 0).

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