A055249 -id:A055249 - cp's OEIS frontend

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

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

A055250 Seventh column of triangle A055249.

Original entry on oeis.org

1, 9, 47, 187, 630, 1898, 5282, 13866, 34831, 84575, 199977, 462973, 1053804, 2365704, 5250660, 11543700, 25177005, 54539205, 117456115, 251676495, 536892146, 1140875254, 2415947382, 5100306062, 10737455195, 22548620283

Offset: 0

Wolfdieter Lang, May 26 2000

Cf. A055249, A035038, partial sums of A034009.

a:= n-> (Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [9,-34,70,-85,61,-24,4][i] else 0 fi)^(n))[1,1]: seq(a(n), n=0..25); # Alois P. Heinz, Aug 05 2008

Table[Sum[(-1)^(n - k) k (-1)^(n - k) Binomial[n + 5, k + 5], {k, 0, n}], {n, 1, 26}] (* Zerinvary Lajos, Jul 08 2009 *)

G.f.: 1/(((1-2*x)^2)*(1-x)^5).
a(n) = A055249(n+6, 6).
For n >= 1, a(n) = A035038(n+6) + Sum_{j=0..n-1} a(j).
a(n) = Sum_{k=0..n+5} Sum_{i=0..n+5} (i-k) * C(n-k+5,i+3). - Wesley Ivan Hurt, Sep 19 2017

A055251 Eighth column of triangle A055249.

Original entry on oeis.org

1, 10, 57, 244, 874, 2772, 8054, 21920, 56751, 141326, 341303, 804276, 1858080, 4223784, 9474444, 21018144, 46195149, 100734354, 218190469, 469866964, 1006759110, 2147634364, 4563581746, 9663887808, 20401343003, 42949963286, 90194651043, 188978952404

Offset: 0

Wolfdieter Lang, May 26 2000

A045618 Partial sums of A000337(n+4),n>=0,
A045889 Partial sums of A045618,
A034009 Partial sums of A045889,
(A055250 Seventh column of triangle A055249) Partial sums of A034009,
(A055251 Eighth column of triangle A055249) Partial sums of A055250. - Vladimir Joseph Stephan Orlovsky, Jul 09 2011

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-43,104,-155,146,-85,28,-4).

Cf. A055249, A035039, partial sums of A055250.

a:= n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [10,-43,104,-155, 146,-85,28,-4][i] else 0 fi)^(n))[1,1]: seq(a(n), n=0..25); # Alois P. Heinz, Aug 05 2008

Table[Sum[(-1)^(n - k) k (-1)^(n - k) Binomial[n + 6, k + 6], {k, 0, n}], {n, 1, 26}] (* Zerinvary Lajos, Jul 08 2009 *)

Vec(1 / ((1 - x)^6*(1 - 2*x)^2) + O(x^30)) \\ Colin Barker, Sep 20 2017

G.f.: 1 / (((1-2*x)^2)*(1-x)^6).
a(n) = A055249(n+7, 7).
For n >= 1, a(n) = A035039(n+7) + Sum_{j=0..n-1} a(j).
a(n) = Sum_{k=0..n+6} Sum_{i=0..n+6} (i-k) * C(n-k+6,i+4). - Wesley Ivan Hurt, Sep 19 2017
a(n) = (1/120)*(38520 - 75*2^(9+n) + 2*(9637 + 15*2^(8+n))*n + 4285*n^2 + 525*n^3 + 35*n^4 + n^5). - Colin Barker, Sep 20 2017

A079282 Diagonal sums of triangle A055249.

Original entry on oeis.org

1, 3, 9, 24, 61, 149, 354, 823, 1881, 4240, 9449, 20857, 45666, 99291, 214589, 461336, 987221, 2103917, 4467394, 9454895, 19951601, 41989408, 88155409, 184670769, 386072386, 805624179, 1678235829, 3490492968, 7249103581, 15034563845

Offset: 0

Paul Barry, Feb 08 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,0,4).

Cf. A055249.

[(n-1)*2^n + Fibonacci(n+3): n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

Table[(n - 1) 2^n + Fibonacci[n + 3], {n, 0, 30}] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{5,-7,0,4},{1,3,9,24},30] (* Harvey P. Dale, May 29 2021 *)

a(n) = sum(sum((k+1)binomial(n-j, n-2j-k), k, 0, n-2j), j, 0, floor(n/2))
a(n) = (n-1)*2^n + F(n+3), with F(n) = A000045(n). - Ralf Stephan, May 09 2004
G.f.: (1-x)^2 / ((1-2*x)^2 * (1-x-x^2)). [Joerg Arndt, Aug 05 2013]

A055248 Triangle of partial row sums of triangle A007318(n,m) (Pascal's triangle). Triangle A008949 read backwards. Riordan (1/(1-2x), x/(1-x)).

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 11, 5, 1, 32, 31, 26, 16, 6, 1, 64, 63, 57, 42, 22, 7, 1, 128, 127, 120, 99, 64, 29, 8, 1, 256, 255, 247, 219, 163, 93, 37, 9, 1, 512, 511, 502, 466, 382, 256, 130, 46, 10, 1, 1024, 1023, 1013, 968, 848, 638, 386, 176, 56, 11, 1

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (also 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/((1-2*z)*(1-x*z/(1-z))).
Binomial transform of the all 1's triangle: as a Riordan array, it factors to give (1/(1-x),x/(1-x))(1/(1-x),x). Viewed as a number square read by antidiagonals, it has T(n,k) = Sum_{j=0..n} binomial(n+k,n-j) and is then the binomial transform of the Whitney square A004070. - Paul Barry, Feb 03 2005
Riordan array (1/(1-2x), x/(1-x)). Antidiagonal sums are A027934(n+1), n >= 0. - Paul Barry, Jan 30 2005; edited by Wolfdieter Lang, Jan 09 2015
Eigensequence of the triangle = A005493: (1, 3, 10, 37, 151, 674, ...); row sums of triangles A011971 and A159573. - Gary W. Adamson, Apr 16 2009
Read as a square array, this is the generalized Riordan array ( 1/(1 - 2*x), 1/(1 - x) ) as defined in the Bala link (p. 5), which factorizes as ( 1/(1 - x), x/(1 - x) )*( 1/(1 - x), x )*( 1, 1 + x ) = P*U*transpose(P), where P denotes Pascal's triangle, A007318, and U is the lower unit triangular array with 1's on or below the main diagonal. - Peter Bala, Jan 13 2016

			The triangle a(n,m) begins:
n\m    0    1    2   3   4   5   6   7  8  9 10 ...
0:     1
1:     2    1
2:     4    3    1
3:     8    7    4   1
4:    16   15   11   5   1
5:    32   31   26  16   6   1
6:    64   63   57  42  22   7   1
7:   128  127  120  99  64  29   8   1
8:   256  255  247 219 163  93  37   9  1
9:   512  511  502 466 382 256 130  46 10  1
10: 1024 1023 1013 968 848 638 386 176 56 11  1
... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
Fourth row polynomial (n=3): p(3,x)= 8 + 7*x + 4*x^2 + x^3.
The matrix inverse starts
   1;
  -2,   1;
   2,  -3,   1;
  -2,   5,  -4,    1;
   2,  -7,   9,   -5,    1;
  -2,   9, -16,   14,   -6,    1;
   2, -11,  25,-  30,   20,   -7,    1;
  -2,  13, -36,   55,  -50,   27,   -8,    1;
   2, -15,  49,  -91,  105,  -77,   35,   -9,  1;
  -2,  17, -64,  140, -196,  182, -112,   44, -10,   1;
   2, -19,  81, -204,  336, -378,  294, -156,  54, -11, 1;
   ...
which may be related to A029653. - _R. J. Mathar_, Mar 29 2013
From _Peter Bala_, Dec 23 2014: (Start)
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/1      \ /1        \ /1       \       /1       \
|2 1     ||0 1       ||0 1      |      |2  1     |
|4 3 1   ||0 2 1     ||0 0 1    |... = |4  5 1   |
|8 7 4 1 ||0 4 3 1   ||0 0 2 1  |      |8 19 9 1 |
|...     ||0 8 7 4 1 ||0 0 4 3 1|      |...      |
|...     ||...       ||...      |      |         |
= A143494. (End)
Matrix factorization of square array as P*U*transpose(P):
/1      \ /1        \ /1 1 1 1 ...\    /1  1  1  1 ...\
|1 1     ||1 1       ||0 1 2 3 ... |   |2  3  4  5 ... |
|1 2 1   ||1 1 1     ||0 0 1 3 ... | = |4  7 11 16 ... |
|1 3 3 1 ||1 1 1 1   ||0 0 0 1 ... |   |8 15 26 42 ... |
|...     ||...       ||...         |   |...            |
- _Peter Bala_, Jan 13 2016

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Peter Bala, Notes on generalized Riordan arrays
Peter Bala, A055248: Rapidly converging series for log(2) and Pi
Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022).
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Norman Lindquist and Gerard Sierksma, Extensions of set partitions, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table I.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

Column sequences: A000079 (powers of 2, m=0), A000225 (m=1), A000295 (m=2), A002662 (m=3), A002663 (m=4), A002664 (m=5), A035038 (m=6), A035039 (m=7), A035040 (m=8), A035041 (m=9), A035042 (m=10).
Row sums: A001792(n) = A055249(n, 0).
Alternating row sums: A011782.
Cf. A011971, A159573. - Gary W. Adamson, Apr 16 2009
Cf. A007318, A008949, A106516, A143494.

a055248 n k = a055248_tabl !! n !! k
a055248_row n = a055248_tabl !! n
a055248_tabl = map reverse a008949_tabl
-- Reinhard Zumkeller, Jun 20 2015

T := (n,k) -> 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n + 1], [n-k + 2], 1/2).
seq(seq(simplify(T(n,k)), k=0..n),n=0..10); # Peter Luschny, Oct 10 2019

a[n_, m_] := Sum[ Binomial[n, m + j], {j, 0, n}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Paul Barry *)
T[n_, k_] := Binomial[n, k] * Hypergeometric2F1[1, k - n, k + 1, -1];
Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* Peter Luschny, Oct 06 2023 *)

a(n, m) = A008949(n, n-m), if n > m >= 0.
a(n, m) = Sum_{k=m..n} A007318(n, k) (partial row sums in columns m).
Column m recursion: a(n, m) = Sum_{j=m..n-1} a(j, m) + A007318(n, m) if n >= m >= 0, a(n, m) := 0 if nG.f. for column m: (1/(1-2*x))*(x/(1-x))^m, m >= 0.
a(n, m) = Sum_{j=0..n} binomial(n, m+j). - Paul Barry, Feb 03 2005
Inverse binomial transform (by columns) of A112626. - Ross La Haye, Dec 31 2006
T(2n,n) = A032443(n). - Philippe Deléham, Sep 16 2009
From Peter Bala, Dec 23 2014: (Start)
Exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(8 + 7*x + 4*x^2/2! + x^3/3!) = 8 + 15*x + 26*x^2/2! + 42*x^3/3! + 64*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).
Let M denote the present triangle. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A143494 (but with a different offset). See the Example section. Cf. A106516. (End)
a(n,m) = Sum_{p=m..n} 2^(n-p)*binomial(p-1,m-1), n >= m >= 0, else 0. - Wolfdieter Lang, Jan 09 2015
T(n, k) = 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n+1], [n-k+2], 1/2). - Peter Luschny, Oct 10 2019
T(n, k) = binomial(n, k)*hypergeom([1, k - n], [k + 1], -1). - Peter Luschny, Oct 06 2023
n-th row polynomial R(n, x) = (2^n - x*(1 + x)^n)/(1 - x). These polynomials can be used to find series acceleration formulas for the constants log(2) and Pi. - Peter Bala, Mar 03 2025

A049611 a(n) = T(n,2), array T as in A049600.

Original entry on oeis.org

0, 1, 4, 13, 38, 104, 272, 688, 1696, 4096, 9728, 22784, 52736, 120832, 274432, 618496, 1384448, 3080192, 6815744, 15007744, 32899072, 71827456, 156237824, 338690048, 731906048, 1577058304, 3388997632, 7264534528, 15535702016

Offset: 0

Clark Kimberling

Refer to A089378 and A075729 for the definition of hierarchies, subhierarchies and one-step transitions. - Thomas Wieder, Feb 28 2004
We may ask for the number of one-step transitions (NOOST) between all unlabeled hierarchies of n elements with the restriction that no subhierarchies are allowed. As an example, consider n = 4 and the hierarchy H1 = [[2,2]] with two elements on level 1 and two on level 2. Starting from H1 the hierarchies [[1, 3]], [[2, 1, 1]], [[1, 2, 1]] can be reached by moving one element only, but [[1, 1, 2]] cannot be reached in a one-step transitition. The solution is n = 1, NOOST = 0; n = 2, NOOST = 1; n = 3, NOOST = 4; n = 4, NOOST = 13; n = 5, NOOST = 38; n = 6, NOOST = 104; n = 7, NOOST = 272; n = 8, NOOST = 688; n = 9, NOOST = 1696. This is sequence A049611. - Thomas Wieder, Feb 28 2004
If X_1,X_2,...,X_n are 2-blocks of a (2n+2)-set X then, for n>=1, a(n+1) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
In each composition (ordered partition) of the integer n, circle the first summand once, circle the second summand twice, etc. a(n) is the total number of circles in all compositions of n (that is, add k*(k+1)/2 for each composition into k parts). Note the O.g.f. is B(A(x)) where A(x)= x/(1-x) and B(x)= x/(1-x)^3.
This is the Riordan transform with the Riordan matrix A097805 (of the associated type) of the triangular number sequence A000217. See a Feb 17 2017 comment on A097805. - Wolfdieter Lang, Feb 17 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Milan Janjic, Two Enumerative Functions
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
S. Kitaev, J. Remmel, p-Ascent Sequences, arXiv:1503.00914 [math.CO], 2015.
Sergey Kitaev, J. B. Remmel, A note on p-Ascent Sequences, Preprint, 2016.
Igor Makhlin, Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties, arXiv:2003.02916 [math.CO], 2020.
Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

a(n+1)= A055252(n, 0), n >= 0. Row sums of triangle A055249.

Cf. A001793, A058396, A075729, A089378, A000217.

CoefficientList[Series[x (1-x)^2/(1-2x)^3,{x,0,40}],x] (* Harvey P. Dale, Sep 24 2013 *)

concat(0, Vec(x*(1-x)^2/(1-2*x)^3+O(x^99))) \\ Charles R Greathouse IV, Jun 12 2015

G.f.: x*(1-x)^2/(1-2*x)^3.
Binomial transform of quarter squares A002620(n+1): a(n) = Sum_{k=0..n} binomial(n, k)*floor((k+1)^2/4). - Paul Barry, May 27 2003
a(n) = 2^(n-4)*(n^2+5*n+2) - 0^n/8. - Paul Barry, Jun 09 2003
a(n+2) = A001787(n+2) + A001788(n). - Creighton Dement, Aug 02 2005
a(n) = Hyper2F1([-n+1, 3], [1], -1) for n>0. - Peter Luschny, Aug 02 2014
a(n) = Sum_{k=0..n-1} Sum_{j=0..n-1} Sum_{i=0..n-1} binomial(n-1, i+j+k). - Yalcin Aktar, Aug 27 2023

A055580 Björner-Welker sequence: 2^n*(n^2 + n + 2) - 1.

Original entry on oeis.org

1, 7, 31, 111, 351, 1023, 2815, 7423, 18943, 47103, 114687, 274431, 647167, 1507327, 3473407, 7929855, 17956863, 40370175, 90177535, 200278015, 442499071, 973078527, 2130706431, 4647288831, 10099884031, 21877489663

Offset: 0

Wolfdieter Lang, May 26 2000; revised Feb 12 2001

a(n) is the d=1 Betti number of the complement of '3-equal' arrangements in n-dimensional real space, see Björner-Welker reference, Table I, pp. 308-309, column '1' with k=3 and Th. 5.2, pp. 297-298.
Binomial transform of [1/2, 2/3, 3/4, 4/5, ...] = 1/2, 7/6, 31/12, 111/20, 351/30, 1023/42, ..., where 2, 6, 12, 20, ... = A002378 (deleting the zero). - Gary W. Adamson, Apr 28 2005
Number of three-dimensional block structures associated with n joint systems in the construction of stable underground structures. - Richard M. Green, Jul 26 2011
Number of monotone mappings from the chain with three points to the complete binary tree of height n (n+1 levels). For example, the seven monotone mappings from the chain with three points (denoted 1,2,3, in order) to the complete binary tree with two levels (with a the root of the tree, and b, c the atoms) are: f(1)=f(2)=f(3)=a; f(1)=f(2)=a, f(3)=b; f(1)=f(2)=a, f(3)=c; f(1)=a, f(2)=f(3)=b; f(1)=a, f(2)=f(3)=c; f(1)=f(2)=f(3)=b; f(1)=f(2)=f(3)=c. - Pietro Codara, Mar 26 2015

H. Barcelo and S. Smith, The discrete fundamental group of the order complex of B_n, Abstract 1020-05-141, 1020th Meeting Amer. Math. Soc., Cincinatti, Ohio, Oct 21-22, 2006.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Henry Adams, Samir Shukla, and Anurag Singh, Čech complexes of hypercube graphs, arXiv:2212.05871 [math.CO], 2022.
H. Barcelo and R. Laubenbacher, Perspectives on A-homotopy theory and its applications, Discr. Math., 298 (2005), 39-61.
H. Barcelo and S. Smith, The discrete fundamental group of the order complex of B_n, arXiv:0711.0915 [math.CO], 2007.
A. Björner and V. Welker, The homology of "k-equal" manifolds and related partition lattices, Adv. Math., 110 (1995), 277-313.
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
G.G. Kocharyan and A.M. Kulyukin, Construction of a three-dimensional block structure on the basis of jointed rock parameters estimating the stability of underground workings, Soil Mech. Found. Eng., 31 (1994), 62-66.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Fourth column of triangle A055252.

Cf. A055252, A055249, A045618, A000337, A001788, A066185.

[2^n*(n^2+n+2)-1: n in [0..35]]; // Vincenzo Librandi, Jul 28 2011

Table[ n*(n+1)*2^(n-2), {n, 0, 26}] // Accumulate // Rest (* Jean-François Alcover, Jul 09 2013, after Paul Barry *)
LinearRecurrence[{7,-18,20,-8},{1,7,31,111},30] (* Harvey P. Dale, Nov 27 2014 *)

a(n)=(n^2+n+2)<Charles R Greathouse IV, Jul 28 2011

a(n) = A055252(n+3, 3).
a(n) = Sum_{j=0..n-1} a(j) + A045618(n), n >= 1.
G.f.: 1/((1-2*x)^3*(1-x)).
Partial sums of A001788 (without leading zero). - Paul Barry, Jun 26 2003
a(n) = A001788(n) - A000337(n). - Jon Perry, Dec 12 2003
a(n) = A119258(n+4,n). - Reinhard Zumkeller, May 11 2006
E.g.f.: 2*(1 + 2*x + 2*x^2)*exp(2*x) - exp(x). - G. C. Greubel, Oct 28 2016
a(n) = Sum_{k=0..n+1} Sum_{i=0..n+1} i^2 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017

Edited (for consistency with change of offset) by M. F. Hasler, Nov 03 2012

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]

A055581 Fifth column of triangle A055252.

Original entry on oeis.org

1, 8, 39, 150, 501, 1524, 4339, 11762, 30705, 77808, 192495, 466926, 1114093, 2621420, 6094827, 14024682, 31981545, 72351720, 162529255, 362807270, 805306341, 1778384868, 3909091299, 8556380130, 18656264161, 40533753824

Offset: 0

Wolfdieter Lang, May 26 2000

a(n) = number of directed column-convex polyominoes of area n+5 having along the lower contour exactly two reentrant corners. - Emeric Deutsch, May 21 2003

Michael De Vlieger, Table of n, a(n) for n = 0..3297
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for linear recurrences with constant coefficients, signature (8, -25, 38, -28, 8).

Cf. A055252, A055249, A045889, partial sums of A055580.

Table[(n^2-n+4)2^(n+1)-7-n,{n,0,30}] (* or *) LinearRecurrence[ {8,-25,38,-28,8},{1,8,39,150,501},30] (* Harvey P. Dale, Nov 07 2011 *)

G.f.: 1/(((1-2*x)^3)*(1-x)^2).
a(n) = A055252(n+4, 4). a(n) = sum(a(j), j=0..n-1)+A045889(n), n >= 1.
a(n) = (n^2-n+4)2^(n+1)-7-n - Emeric Deutsch, May 21 2003
a(0)=1, a(1)=8, a(2)=39, a(3)=150, a(4)=501, a(n) = 8*a(n-1)- 25*a(n-2)+ 38*a(n-3)-28*a(n-4)+8*a(n-5). [Harvey P. Dale, Nov 07 2011]

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

Original entry on oeis.org

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

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

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A055250 Seventh column of triangle A055249.

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A055251 Eighth column of triangle A055249.

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A079282 Diagonal sums of triangle A055249.

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A055248 Triangle of partial row sums of triangle A007318(n,m) (Pascal's triangle). Triangle A008949 read backwards. Riordan (1/(1-2x), x/(1-x)).

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A049611 a(n) = T(n,2), array T as in A049600.

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A055580 Björner-Welker sequence: 2^n*(n^2 + n + 2) - 1.

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