A005774 -id:A005774 - cp's OEIS frontend

A038622 Triangular array that counts rooted polyominoes.

1, 2, 1, 5, 3, 1, 13, 9, 4, 1, 35, 26, 14, 5, 1, 96, 75, 45, 20, 6, 1, 267, 216, 140, 71, 27, 7, 1, 750, 623, 427, 238, 105, 35, 8, 1, 2123, 1800, 1288, 770, 378, 148, 44, 9, 1, 6046, 5211, 3858, 2436, 1296, 570, 201, 54, 10, 1, 17303, 15115, 11505, 7590, 4302, 2067, 825, 265

Offset: 0

N. J. A. Sloane, torsten.sillke(AT)lhsystems.com

The PARI program gives any row k and any n-th term for this triangular array in square or right triangle array format. - Randall L Rathbun, Jan 20 2002
Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Triangle read by rows = partial sums of A064189 terms starting from the right. - Gary W. Adamson, Oct 25 2008
Column k has e.g.f. exp(x)*(Bessel_I(k,2x)+Bessel_I(k+1,2x)). - Paul Barry, Mar 08 2011

			From _Paul Barry_, Mar 08 2011: (Start)
Triangle begins
     1;
     2,    1;
     5,    3,    1;
    13,    9,    4,   1;
    35,   26,   14,   5,   1;
    96,   75,   45,  20,   6,   1;
   267,  216,  140,  71,  27,   7,  1;
   750,  623,  427, 238, 105,  35,  8, 1;
  2123, 1800, 1288, 770, 378, 148, 44, 9, 1;
Production matrix is
  2, 1,
  1, 1, 1,
  0, 1, 1, 1,
  0, 0, 1, 1, 1,
  0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 1, 1
(End)

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357. See Table 1 on page 340.

Cf. A005773 (1st column), A005774 (2nd column), A005775, A066822, A000244 (row sums).

Cf. A064189, A007318, A061554, A092392.

import Data.List (transpose)
a038622 n k = a038622_tabl !! n !! k
a038622_row n = a038622_tabl !! n
a038622_tabl = iterate (\row -> map sum $
   transpose [tail row ++ [0,0], row ++ [0], [head row] ++ row]) [1]
-- Reinhard Zumkeller, Feb 26 2013

T := (n,k) -> simplify(GegenbauerC(n-k,-n+1,-1/2)+GegenbauerC(n-k-1,-n+1,-1/2)):
for n from 1 to 9 do seq(T(n,k),k=1..n) od; # Peter Luschny, May 12 2016

nmax = 10; t[n_ /; n > 0, k_ /; k >= 1] := t[n, k] = t[n-1, k-1] + t[n-1, k] + t[n-1, k+1]; t[0, 0] = 1; t[0, ] = 0; t[?Negative, ?Negative] = 0; t[n, 0] := 2 t[n-1, 0] + t[n-1, 1]; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]](* Jean-François Alcover, Nov 09 2011 *)

s=[0,1]; {A038622(n,k)=if(n==0,1,t=(2*(n+k)*(n+k-1)*s[2]+3*(n+k-1)*(n+k-2)*s[1])/((n+2*k-1)*n); s[1]=s[2]; s[2]=t; t)}

a(n, k) = a(n-1, k-1) + a(n-1, k) + a(n-1, k+1) for k>0, a(n, k) = 2*a(n-1, k) + a(n-1, k+1) for k=0.
Riordan array ((sqrt(1-2x-3x^2)+3x-1)/(2x(1-3x)),(1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array ((1-x)/(1+x+x^2),x/(1+x+x^2)). First column is A005773(n+1). Row sums are 3^n (A000244). If L=A038622, then L*L' is the Hankel matrix for A005773(n+1), where L' is the transpose of L. - Paul Barry, Sep 18 2006
T(n,k) = GegenbauerC(n-k,-n+1,-1/2) + GegenbauerC(n-k-1,-n+1,-1/2). In this form also the missing first column of the triangle 1,1,1,3,7,19,... (cf. A002426) can be computed. - Peter Luschny, May 12 2016
From Peter Bala, Jul 12 2021: (Start)
T(n,k) = Sum_{j = k..n} binomial(n,j)*binomial(j,floor((j-k)/2)).
Matrix product of Riordan arrays ( 1/(1 - x), x/(1 - x) ) * ( (1 - x*c(x^2))/(1 - 2*x), x*c(x^2) ) = A007318 * A061554 (triangle version), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
Triangle equals A007318^(-1) * A092392 * A007318. (End)
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + x)*(1 + x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

More terms from David W. Wilson

A026323 Irregular triangular array T read by rows: T(0,0) = 1, T(0,1) = T(0,2) = 0; T(1,0) = T(1,1) = T(1,2) = 1, T(1,3) = 0; for n >= 2, T(n,0) = 1, T(n,1) = T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 2,3,...,n+1 and T(n,n+2) = T(n-1,n) + T(n-1,n+1).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 0, 1, 2, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 4, 10, 16, 19, 16, 9, 1, 5, 15, 30, 45, 51, 44, 25, 1, 6, 21, 50, 90, 126, 140, 120, 69, 1, 7, 28, 77, 161, 266, 356, 386, 329, 189, 1, 8, 36, 112, 266, 504, 783, 1008, 1071, 904, 518, 1, 9, 45, 156, 414, 882, 1553, 2295, 2862, 2983, 2493

Offset: 1

Clark Kimberling

			First five rows:
  1  0  0
  1  1  1  0
  1  2  3  2  1
  1  3  6  7  6  3
  1  4 10 16 19 16  0

Cf. A005774, A027907.

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 0] = 1; t[1, 1] = 1;
t[1, 2] = 1; t[1, 3] = 0;
t[n_, 0] := 1; t[n_, 1] := t[n - 1, 0] + t[n - 1, 1];
t[n_, k_] := t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k] /; 2 <= k && k <= n + 1;
t[n_, k_] := t[n - 1, k - 2] + t[n - 1, k - 1] /; k == n + 2;
u = Table[t[n, k], {n, 0, 12}, {k, 0, n + 2}];
TableForm[u] (*A026323 array*)
v = Flatten[u] (*A026323 sequence*)
(* Clark Kimberling, Aug 21 2014 *)

Beginning of sequence changed from 1,0,0,1,1,1,1,2,3,2,1,1,3,6,7,6,3 by N. J. A. Sloane (7/98).

A026010 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2. Also a(n) = sum of numbers in row n+1 of array T defined in A026009.

Original entry on oeis.org

1, 2, 4, 7, 14, 25, 50, 91, 182, 336, 672, 1254, 2508, 4719, 9438, 17875, 35750, 68068, 136136, 260338, 520676, 999362, 1998724, 3848222, 7696444, 14858000, 29716000, 57500460, 115000920, 222981435, 445962870, 866262915, 1732525830, 3370764540

Offset: 0

Clark Kimberling

nonn

Conjecture: a(n) is the number of integer compositions of n + 2 in which the even parts appear as often at even positions as at odd positions (confirmed up to n = 19). - Gus Wiseman, Mar 17 2018

			The a(3) = 7 compositions of 5 in which the even parts appear as often at even positions as at odd positions are (5), (311), (131), (113), (221), (122), (11111). Missing are (41), (14), (32), (23), (212), (2111), (1211), (1121), (1112). - _Gus Wiseman_, Mar 17 2018

Michael De Vlieger, Table of n, a(n) for n = 0..3326
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.

First differences of A050168. Pairwise sums of A037952.

Cf. A000712, A001405, A005774, A045931, A063886, A097613, A130780, A171966, A239241, A299926, A300061, A300787, A300788, A300789.

[(&+[Binomial(Floor((n+k)/2), Floor(k/2)): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Nov 08 2018

Array[Sum[Binomial[Floor[(# + k)/2], Floor[k/2]], {k, 0, #}] &, 34, 0] (* Michael De Vlieger, May 16 2018 *)
Table[2^(-1 + n)*(((2 + 3*#)*Gamma[(1 + #)/2])/(Sqrt[Pi]*Gamma[2 + #/2]) &[n + Mod[n, 2]]), {n,0,40}] (* Peter Pein, Nov 08 2018 *)
Table[(1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*CatalanNumber[(2*n + 1 - (-1)^n)/4], {n, 0, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, n--; sum(k=0,n, binomial(floor((n+k)/2), floor(k/2)))) \\ G. C. Greubel, Nov 08 2018

a(2*n) = ((3*n + 1)/(2*n + 1))*C(2*n + 1, n)= A051924(1+n), n>=0, a(2*n-1) = a(2*n)/2 = A097613(1+n), n >= 1. - Herbert Kociemba, May 08 2004
a(n) = Sum_{k=0..n} binomial(floor((n+k)/2), floor(k/2)). - Paul Barry, Jul 15 2004
Inverse binomial transform of A005774: (1, 3, 9, 26, 75, 216, ...). - Gary W. Adamson, Oct 22 2007
Conjecture: (n+3)*a(n) - 2*a(n-1) + (-5*n-3)*a(n-2) + 2*a(n-3) + 4*(n-3)*a(n-4) = 0. - R. J. Mathar, Jun 20 2013
a(n) = (1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*Catalan((2*n + 1 - (-1)^n)/4), where Catalan is the Catalan number = A000108. - G. C. Greubel, Nov 08 2018

A114422 Riordan array (1/sqrt(1-2x-3x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 7, 9, 5, 1, 19, 26, 19, 7, 1, 51, 75, 65, 33, 9, 1, 141, 216, 211, 132, 51, 11, 1, 393, 623, 665, 483, 235, 73, 13, 1, 1107, 1800, 2058, 1674, 963, 382, 99, 15, 1, 3139, 5211, 6294, 5598, 3663, 1739, 581, 129, 17, 1, 8953, 15115, 19095, 18261, 13243

Offset: 0

Paul Barry, Feb 12 2006

First column is central trinomial numbers A002426.
Second column is A005774.
Third column is A025568.
Row sums are A116387.
Diagonal sums are A116388.
Product of A007318 and A116382.
Column k has e.g.f. exp(x)*Sum_{j=0..k} C(k,j)*Bessel_I(k+j,2*x).

			Triangle begins
1,
1, 1,
3, 3, 1,
7, 9, 5, 1,
19, 26, 19, 7, 1,
51, 75, 65, 33, 9, 1,
141, 216, 211, 132, 51, 11, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T:=Flat(List([0..10], n->List([0..n], k->Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j))))); # G. C. Greubel, Dec 15 2018

[[(&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Dec 15 2018

T[n_, k_] := Sum[Binomial[n, j - k]*Binomial[j, n - j], {j,0,n}]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

{T(n,k) = sum(j=0,n, binomial(n, j-k)*binomial(j, n-j))};
for(n=0, 10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 15 2018

[[sum(binomial(n, j-k)*binomial(j, n-j) for j in range(n+1)) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Dec 15 2018

Riordan array (1/sqrt(1-2*x-3*x^2), (1-x-2*x^2-sqrt(1-2*x-3*x^2) ) / (2*x^2)).

Number triangle T(n,k) = Sum_{j=0..n} C(n,j-k)*C(j,n-j).

A005775 Number of compact-rooted directed animals of size n having 3 source points.

Original entry on oeis.org

1, 4, 14, 45, 140, 427, 1288, 3858, 11505, 34210, 101530, 300950, 891345, 2638650, 7809000, 23107488, 68375547, 202336092, 598817490, 1772479905, 5247421410, 15538054455, 46019183840, 136325212750, 403933918375, 1197131976846, 3548715207534, 10521965227669

Offset: 3

Simon Plouffe

Binomial transform of A037955. - Paul Barry, Dec 28 2006
Apparently, the number of Dyck paths of semilength n that contain at least one UUU but avoid UUU's starting above level 0. - David Scambler, Jul 02 2013
a(n) = number of paths in the half-plane x >= 0 from (0,0) to (n-1,2) or (n-1,-3), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=5, we have the 14 paths: HHUU, UUHH, UHHU, HUUH, HUHU, UHUH, UDUU, UUDU, UUUD, DUUU, DDDH, HDDD, DHDD, DDHD. - José Luis Ramírez Ramírez, Apr 19 2015

			G.f. = x^3 + 4*x^4 + 14*x^5 + 45*x^6 + 140*x^7 + 427*x^8 + 1288*x^9 + 3858*x^10 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.

Cf. A005773.
k=2 column of array in A038622.
Cf. A005774, A066822.

a005775 = flip a038622 2 . (subtract 1)  -- Reinhard Zumkeller, Feb 26 2013

seq(simplify(GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2)),n=3..28); # Peter Luschny, May 12 2016

nmax = 28; t[n_ /; n > 0, k_ /; k >= 1] := t[n, k] = t[n-1, k-1] + t[n-1, k] + t[n-1, k+1]; t[0, 0] = 1; t[0, ] = 0; t[?Negative, ?Negative] = 0; t[n, 0] := 2*t[n-1, 0] + t[n-1, 1]; a[n_] := t[n-1, 2]; Table[a[n], {n, 3, nmax} ] (* Jean-François Alcover, Jul 03 2013, from A038622 *)

{a(n) = polcoeff( (x^2 + x - 1 + (x^2 - 3*x + 1) * sqrt((1 + x) / (1 - 3*x) + x^3 * O(x^n))) / (2*x^2), n)};

{a(n) = n--; sum(k=0, n, binomial(n, k) * binomial(k, k\2 -1))}; /* Michael Somos, May 12 2016 */

D-finite with recurrence (n+2)*(n-3)*a(n) = 2*n*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2), a(2)=0, a(3)=1. - Michael Somos, Feb 02 2002
G.f.: (x^2 + x - 1 +(x^2 - 3*x + 1)*sqrt((1+x)/(1-3*x)))/(2*x^2).
From Paul Barry, Dec 28 2006: (Start)
E.g.f.: exp(x)*(Bessel_I(2,2*x) + Bessel_I(3,2*x));
a(n+1) = Sum_{k=0..n} C(n,k)*C(k,floor(k/2)-1). (End)
a(n) ~ 3^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 25 2014
G.f.: (z^3*M(z)^2+z^4*M(z)^3)/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) = GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2). - Peter Luschny, May 12 2016
0 = a(n)*(+9*a(n+1) - 63*a(n+2) - 54*a(n+3) + 87*a(n+4) - 21*a(n+5))+ a(n+1)*(+21*a(n+1) + 79*a(n+2) + 13*a(n+3) - 118*a(n+4) + 35*a(n+5)) + a(n+2)*(-14*a(n+2) + 79*a(n+3) - 67*a(n+4) + 14*a(n+5)) + a(n+3)*(+6*a(n+3) + 19*a(n+4) - 11*a(n+5)) + a(n+4)*(+a(n+4) + a(n+5)) if n >= 0. - Michael Somos, May 12 2016
a(n) = A005773(n) - A001006(n) for n >= 3. - John Keith, Nov 20 2020

More terms from Randall L Rathbun, Jan 19 2002

Edited by Michael Somos, Feb 02 2002

A066822 The fourth column of A038622, triangular array that counts rooted polyominoes.

Original entry on oeis.org

1, 5, 20, 71, 238, 770, 2436, 7590, 23397, 71566, 217646, 659022, 1988805, 5986176, 17980968, 53922096, 161492571, 483149385, 1444245936, 4314214443, 12880107548, 38436170366, 114657076900, 341926185770, 1019435748435, 3038815305981, 9056974493700

Offset: 0

Randall L Rathbun, Jan 19 2002

There is a general solution for all rows of this triangular array: For the k-th row and n-th term on this row: a(0)=0; a(1)=1; a(n) = (2*k-1+n)*n*a(n) = 2*(n+k)*(n+k-1)*a(n-1) + 3*(n+k-1)*(n+k-2)*a(n-2).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.

Cf. A038622.

Cf. A005773, A005774, A005775.

a066822 = flip a038622 3 . (+ 3)  -- Reinhard Zumkeller, Feb 26 2013

a := n -> simplify(GegenbauerC(n,-n+1-4,-1/2)+GegenbauerC(n-1,-n-3,-1/2)):
seq(a(n), n=0..20); # Peter Luschny, May 12 2016

Table[GegenbauerC[n,-n-3,-1/2]+GegenbauerC[n-1,-n-3,-1/2],{n,0,40}] (* Harvey P. Dale, Feb 20 2017 *)

s=[0,1]; {A038622(n,k)=if(n==0,1,t=(2*(n+k)*(n+k-1)*s[2]+3*(n+k-1)*(n+k-2)*s[1])/((n+2*k-1)*n); s[1]=s[2]; s[2]=t; t)}

a(0)=0; a(1)=1; (n+7)*n*a(n)=2*(n+4)*(n+3)*a(n-1) + 3*(n+3)*(n+2)*a(n-2).
a(n) = ((-3)^(1/2)/9)*(-2*(n+7)^(-1)*(n+4)*(-1)^n*hypergeom([3/2, n+6],[2],4/3)-(n+6)^(-1)*(-1)^n*(5*n+18)*hypergeom([3/2, n+5],[2],4/3)). - Mark van Hoeij, Oct 31 2011
a(n) ~ 3^(n+7/2) / sqrt(Pi*n). - Vaclav Kotesovec, Mar 10 2014
a(n) = GegenbauerC(n,-n+1-4,-1/2)+GegenbauerC(n-1,-n-3,-1/2). - Peter Luschny, May 12 2016

More terms from Harvey P. Dale, Feb 20 2017

A035045 Inverse binomial transform of A002054.

Original entry on oeis.org

1, 4, 12, 35, 101, 291, 839, 2423, 7011, 20326, 59038, 171777, 500603, 1461032, 4269828, 12493857, 36599403, 107325540, 315027276, 925501857, 2721208599, 8007114171, 23577440439, 69470880381, 204821487269, 604223501426, 1783419354954, 5266582196407, 15560042628205

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A005774.

Table[Sum[(-1)^(n-k)*Binomial[n,k]*Binomial[2k+3,k],{k,0,n}],{n,0,22}] (* Vaclav Kotesovec, Oct 08 2012 *)

a(n)=sum(k=0,n,(-1)^(n-k)*binomial(n,k)*binomial(2*k+3,k) ); \\ Joerg Arndt, May 04 2013

Recurrence: (n+3)*(3*n-1)*a(n) = (6*n^2+19*n+7)*a(n-1) + 3*(n-1)*(3*n+2)*a(n-2). - Vaclav Kotesovec, Oct 08 2012

a(n) ~ 4*3^(n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 08 2012

A098494 Triangle read by rows: coefficients of polynomials E(n,x) related to partitions with parts occurring at most thrice.

Original entry on oeis.org

1, 1, -1, 1, -5, 4, 1, -12, 35, -30, 1, -22, 143, -362, 312, 1, -35, 405, -2065, 4814, -4200, 1, -51, 925, -7965, 35434, -78744, 69120, 1, -70, 1834, -24010, 173929, -709240, 1525236, -1345680, 1, -92, 3290, -61040, 655529, -4235588, 16255420, -34148400, 30240000

Offset: 0

Ralf Stephan, Sep 12 2004

The polynomials generate (-1)^k*n! times the diagonals of A098493.

			E(0,x) = 1
E(1,x) = x - 1
E(2,x) = x^2 - 5*x + 4
E(3,x) = x^3 - 12*x^2 + 35*x - 30
E(4,x) = x^4 - 22*x^3 + 143*x^2 - 362*x + 312
E(5,x) = x^5 - 35*x^4 + 405*x^3 - 2065*x^2 + 4814*x - 4200

Seiichi Manyama, Rows n = 0..139, flattened
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Columns include -A000326.
Constant terms E(n, 0) = -E(n-1, -1) = n!/2*A085455 = (-1)^n*n!*A005773.
Row sums are E(n, 1) = (-1)^n*n!*A005774(n-2). [corrected by Seiichi Manyama, Feb 04 2023]

E(n+1,x+1) - E(n+1,x) = (n+1) * ( E(n,x) - n * E(n-1,x-1) ).

A383527 Partial sums of A005773.

Original entry on oeis.org

1, 2, 4, 9, 22, 57, 153, 420, 1170, 3293, 9339, 26642, 76363, 219728, 634312, 1836229, 5328346, 15494125, 45137995, 131712826, 384900937, 1126265986, 3299509114, 9676690939, 28407473191, 83470059532, 245465090758, 722406781935, 2127562036990, 6270020029353

Offset: 0

Mélika Tebni, Apr 29 2025

For p prime of the form 4*k+3 (A002145), a(p) == 0 (mod p).
For p Pythagorean prime (A002144), a(p) - 2 == 0 (mod p).
a(n) (mod 2) = A010059(n).
a(A000069(n+1)) is even.
a(A001969(n+1)) is odd.

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A000069, A001969, A002144, A002145, A005773, A005774, A005775, A010059, A097893, A122896, A167630, A210736, A211278, A257520.

gf := (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)):
a := n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n = 0 .. 29);
# Recurrence:
a:= proc(n) option remember; `if`(n<=2, 2^n, 3*a(n-1) - (6/n-1)*a(n-2) + (6/n-3)*a(n-3)) end:
seq(a(n), n = 0 .. 29);

Module[{a, n}, RecurrenceTable[{a[n] == 3*a[n-1] - (6-n)*a[n-2]/n + 3*(2-n)*a[n-3]/n, a[0] == 1, a[1] == 2, a[2] == 4}, a, {n, 0, 30}]] (* Paolo Xausa, May 05 2025 *)

from math import comb as C
def a(n):
  return sum(C(n, k)*abs(sum((-1)**j*C(k, j) for j in range(k//2 + 1))) for k in range(n + 1))
print([a(n) for n in range(30)])

First differences of A211278.
a(n) = Sum_{k=0..n} A167630(n, k).
Binomial transform of A210736 (see Python program).
G.f.: (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)).
E.g.f.: (Integral_{x=-oo..oo} BesselI(0,2*x) dx + (1 + BesselI(0,2*x)) / 2)*exp(x).
Recurrence: n*a(n) = 3*n*a(n-1) - (6-n)*a(n-2) + 3*(2-n)*a(n-3). If n <= 2, a(n) = 2^n.
a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, May 02 2025
From Mélika Tebni, May 09 2025: (Start)
a(n) = A257520(n) + A097893(n-1) for n > 0.
a(n) = Sum_{j=0..n}(Sum_{k=0..j} A122896(j, k)).
a(n+2) - 3*a(n+1) + 2*a(n) = A005774(n).
a(n+2) - 4*a(n+1) + 4*a(n) - a(n-1) = A005775(n) for n >= 3. (End)

A114580 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k ascents (0<=k<=floor(n/2)); an ascent is a maximal string of upsteps.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 14, 6, 1, 26, 23, 1, 1, 46, 70, 10, 1, 79, 186, 56, 1, 1, 133, 451, 235, 15, 1, 221, 1025, 825, 115, 1, 1, 364, 2220, 2562, 630, 21, 1, 596, 4634, 7274, 2794, 211, 1, 1, 972, 9396, 19286, 10696, 1456, 28, 1, 1581, 18612, 48450, 36715

Offset: 0

Emeric Deutsch, Dec 09 2005

Row n contains 1+floor(n/2) terms. Row sums are the Motzkin numbers (A001006). Sum(k*T(n,k),k=0..floor(n/2))=A005774(n-1).

			T(4,1) = 7 because we have HH(U)D, H(U)DH, H(U)HD, (U)DHH, (U)HDH, (U)HHD and (UU)HH, where U=(1,1), H=(1,0), D=(1,-1) (the ascents are shown between parentheses).
Triangle begins:
1;
1;
1,  1;
1,  3;
1,  7,  1;
1, 14,  6;
1, 26, 23, 1;

Alois P. Heinz, Rows n = 0..200, flattened
Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2.

Cf. A001006, A005774.

G:=1/2*(1-z+z^2-t*z^2-sqrt(1-z^2-2*z+2*z^3-2*z^3*t-2*z^2*t+z^4-2*z^4*t+z^4*t^2))/z^2/(z*t+1-z): Gserz:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(Gserz,z^n)) od: for n from 0 to 15 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, expand(`if`(t, 1, z)*b(x-1, y-1, true)
      +b(x-1, y+1, false)+b(x-1, y, false))))
    end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, false)):
seq(T(n), n=0..20);  # Alois P. Heinz, Mar 11 2014

b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[If[t, 1, z]*b[x-1, y-1, True] + b[x-1, y+1, False] + b[x-1, y, False]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, False]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

G.f.: G(t,z) satisfies G = 1+zG+z^2[t(1+zG)+G-1-zG]G.

cp's OEIS Frontend

A038622 Triangular array that counts rooted polyominoes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A026323 Irregular triangular array T read by rows: T(0,0) = 1, T(0,1) = T(0,2) = 0; T(1,0) = T(1,1) = T(1,2) = 1, T(1,3) = 0; for n >= 2, T(n,0) = 1, T(n,1) = T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 2,3,...,n+1 and T(n,n+2) = T(n-1,n) + T(n-1,n+1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A026010 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2. Also a(n) = sum of numbers in row n+1 of array T defined in A026009.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A114422 Riordan array (1/sqrt(1-2*x-3*x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.

Views

Author

Keywords

Comments

Examples

Links

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A005775 Number of compact-rooted directed animals of size n having 3 source points.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

Extensions

A066822 The fourth column of A038622, triangular array that counts rooted polyominoes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A114422 Riordan array (1/sqrt(1-2x-3x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.