A030267 -id:A030267 - cp's OEIS frontend

A001870 Expansion of (1-x)/(1 - 3*x + x^2)^2.

1, 5, 19, 65, 210, 654, 1985, 5911, 17345, 50305, 144516, 411900, 1166209, 3283145, 9197455, 25655489, 71293590, 197452746, 545222465, 1501460635, 4124739581, 11306252545, 30928921224, 84451726200, 230204999425

Offset: 0

N. J. A. Sloane

a(n) = ((n+1)*F(2*n+3)+(2*n+3)*F(2*(n+1)))/5 with F(n)=A000045(n) (Fibonacci numbers). One half of odd-indexed A001629(n), n >= 2 (Fibonacci convolution).
Convolution of F(2n+1) (A001519) and F(2n+2) (A001906(n+1)). - Graeme McRae, Jun 07 2006
Number of reentrant corners along the lower contours of all directed column-convex polyominoes of area n+3 (a reentrant corner along the lower contour is a vertical step that is followed by a horizontal step). a(n) = Sum_{k=0..ceiling((n+1)/2)} k*A121466(n+3,k). - Emeric Deutsch, Aug 02 2006
From Wolfdieter Lang, Jan 02 2012: (Start)
a(n) = A024458(2*n), n >= 1 (bisection, even arguments).
a(n) is also the odd part of the bisection of the half-convolution of the sequence A000045(n+1), n >= 0, with itself. See a comment on A201204 for the definition of the half-convolution of a sequence with itself. There one also finds the rule for the o.g.f. which in this case is Chato(x)/2 with the o.g.f. Chato(x) = 2*(1-x)/(1-3*x+x^2)^2 of A001629(2*n+3), n >= 0.
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Elena Barcucci, Renzo Pinzani, and Renzo Sprugnoli , Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Éva Czabarka, Rigoberto Flórez, and Leandro Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Éva Czabarka, Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math. 341 (10) (2018), 2789-2807. See Cor. 6.
Emeric Deutsch and Helmut 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. 5-6, 14-15, 17, 19.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
Pieter Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
John Riordan, Notes to N. J. A. Sloane, Jul. 1968
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

a(n) = A060921(n+1, 1)/2.
Partial sums of A030267. First differences of A001871.
Cf. A121466.
Cf. A023610.

F:=Fibonacci;; List([0..30], n-> ((n+1)*F(2*n+3)+(2*n+3)*F(2*(n+1)))/5); # G. C. Greubel, Jul 15 2019

a001870 n = a001870_list !! n
a001870_list = uncurry c $ splitAt 1 $ tail a000045_list where
   c us vs'@(v:vs) = (sum $ zipWith (*) us vs') : c (v:us) vs
-- Reinhard Zumkeller, Oct 31 2013

I:=[1, 5, 19, 65]; [n le 4 select I[n] else 6*Self(n-1) -11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 10 2012

A001870:=-(-1+z)/(z**2-3*z+1)**2; # Simon Plouffe in his 1992 dissertation.

CoefficientList[Series[(1-x)/(1-3*x+x^2)^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 10 2012 *)
LinearRecurrence[{6,-11,6,-1},{1,5,19,65},30] (* Harvey P. Dale, Aug 17 2013 *)
With[{F=Fibonacci}, Table[((n+1)*F[2*n+3]+(2*n+3)*F[2*n+2])/5, {n,0,30}]] (* G. C. Greubel, Jul 15 2019 *)

Vec((1-x)/(1-3*x+x^2)^2+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

f=fibonacci; [((n+1)*f(2*n+3)+(2*n+3)*f(2*n+2))/5 for n in (0..30)] # G. C. Greubel, Jul 15 2019

a(n) = Sum_{k=1..n+1} k*binomial(n+k+1, 2k). - Emeric Deutsch, Jun 11 2003
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 10 2012
a(n) = (A238846(n) + A001871(n))/2. - Philippe Deléham, Mar 06 2014
a(n) = ((2*n-1)*Fibonacci(2*n) - n*Fibonacci(2*n-1))/5 [Czabarka et al.]. - N. J. A. Sloane, Sep 18 2018
E.g.f.: exp(3*x/2)*(5*(5 + 11*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(13 + 25*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025

More terms from Christian G. Bower

A038731 Number of columns in all directed column-convex polyominoes of area n+1.

Original entry on oeis.org

1, 3, 10, 32, 99, 299, 887, 2595, 7508, 21526, 61251, 173173, 486925, 1362627, 3797374, 10543724, 29180067, 80521055, 221610563, 608468451, 1667040776, 4558234018, 12441155715, 33900136297, 92230468249, 250570010499, 679844574322, 1842280003640

Offset: 0

Clark Kimberling, May 02 2000

Apply Riordan array (1/(1-x), x/(1-x)^2) to n+1. - Paul Barry, Oct 13 2009

Binomial transform of (A001629 shifted left twice). - R. J. Mathar, Feb 06 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Elena Barcucci, Renzo Pinzani, and Renzo Sprugnoli , Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
Éva Czabarka, Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumeration of peaks and valleys on non-decreasing Dyck paths, Disc. Math. 341(10) (2018), 2789-2807; see Theorem 2 on p. 2791.
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. 5, 15, 17, 19.
Jesus Salas and Alan D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys. 135 (2009) 279-373, arXiv:0711.1738. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Row-sums of array T as in A038730.
First differences of A030267.
Row sums of A318942(n+1).
Cf. A000045.

a038731 n = a038731_list !! n
a038731_list = c [1] $ tail a000045_list where
   c us vs'@(v:vs) = (sum $ zipWith (*) us vs') : c (v:us) vs
-- Reinhard Zumkeller, Oct 31 2013

I:=[1, 3, 10, 32]; [n le 4 select I[n] else 6*Self(n-1)-11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 04 2012

Table[Sum[Binomial[n, k]*CoefficientList[Series[1/(1 - x - x^2)^2, {x, 0, k}], x][[-1]], {k, 0, n}], {n, 0, 27}] (* Arkadiusz Wesolowski, Feb 03 2012 *)
LinearRecurrence[{6, -11, 6, -1}, {1, 3, 10, 32}, 30] (* Vincenzo Librandi, Feb 04 2012 *)

5*a(n) = (2n+1)*F(2n+2) - (n-4)*F(2n+1), where the F(n)'s are the Fibonacci numbers, F(0)=0, F(1)=1.
a(n) = Sum_{k=1..n+1} k*binomial(n+k-1, 2k-2). - Emeric Deutsch, Jun 11 2003
From Paul Barry, Oct 13 2009: (Start)
G.f.: (1-x)^3/(1-3x+x^2)^2.
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(k+1). (End)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - R. J. Mathar, Feb 06 2010
a(n) = Sum_{k=0..n} (F(2k)+0^k)*F(2n-2k+1). - Paul Barry, Jun 23 2010
E.g.f.: exp(3*x/2)*(5*(5 + 4*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(7 + 10*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025

Entry improved by comments from Emeric Deutsch, Jun 14 2001

A153294 G.f.: A(x) = F(xF(x)^2) = F(F(x)-1) where F(x) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan).

Original entry on oeis.org

1, 1, 4, 18, 86, 427, 2180, 11373, 60380, 325259, 1773842, 9776637, 54380144, 304905223, 1721650832, 9782051362, 55888463214, 320898932595, 1850762866662, 10717217871255, 62287285235230, 363212668363520, 2124430957852380

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

Ignoring a(0), the series reversal gives A030267 with alternating signs: 1, -4, 14, -46, 145, -444, ... - Vladimir Reshetnikov, Aug 03 2019

			G.f.: A(x) = F(x*F(x)^2) = 1 + x + 4*x^2 + 18*x^3 + 86*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 44*x^3 + 224*x^4 + 1170*x^5 + 6226*x^6 +...
F(x)^2*A(x)^2 = 1 + 4*x + 18*x^2 + 86*x^3 + 427*x^4 + 2180*x^5 +...
From _Peter Bala_, Jul 21 2015: (Start)
Let B(x) = (A(x) - 1)/x = Sum_{n >= 0} a(n+1)*x^n. Then 1 + x*B'(x)/B (x)  = 1 + 4*x + 20*x^2 + 106*x^3 + ... is the o.g.f. for A243585.
x*sqrt(B(x)) = x + 2*x^2 + 7*x^3 + 29*x^4 + ... is the o.g.f for A007582. (End)

Cf. A001764, A000108; A030267, A153293, A153295, A007852, A007856, A243585.

a[0] = 1; a[n_] := 2^(n-1) (2n-1)!! ((Hypergeometric2F1[-1/2, -n-1, n, -4] - 1)/(n+1)! + 2 Hypergeometric2F1[1/2, -n, n+1, -4]/(n n!)); Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)
Flatten[{1, Table[Sum[Binomial[2*k + 1, k]/(2*k + 1)*Binomial[2*(n-k) + 2*k, n-k]*2*k/(2*(n-k) + 2*k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 07 2015 *)
A153294[0] := 1; A153294[n_] := (A243585[n] - A007856[n+1])/n;
Table[A153294[n], {n, 0, 22}] (* Peter Luschny, Aug 04 2019 *)

{a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*binomial(2*(n-k)+2*k,n-k)*2*k/(2*(n-k)+2*k)))}

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(2n,n-k)*k/n for n>0 with a(0)=1.
G.f.: A(x) = [1 - sqrt(5 - 4*F(x))]/(2*F(x)-2) where F(x) = (1-sqrt(1-4x))/(2x).
G.f. satisfies: A(x) = 1 + x*F(x)^2*A(x)^2 where F(x) is the g.f. of A000108.
G.f. satisfies: A(x*G(x)) = F(x*G(x)^3) = F(G(x)-1) where G(x) = F(x*G(x)) is the g.f. of A001764 and F(x) is the g.f. of A000108.
For n > 0, a(n) = 2^(n-1)*(2*n-1)!!*((hypergeom([-1/2,-n-1], [n], -4) - 1)/(n+1)! + 2*hypergeom([1/2,-n], [n+1], -4)/(n*n!)). - Vladimir Reshetnikov, Nov 07 2015
a(n) ~ 5^(2*n + 1/2) / (sqrt(3*Pi) * n^(3/2) * 4^n). - Vaclav Kotesovec, Nov 07 2015
a(n) = (A243585(n) - A007856(n+1)) / n for n >= 1. - Peter Luschny, Aug 04 2019

A030279 COMPOSE squares with squares.

Original entry on oeis.org

1, 8, 50, 276, 1397, 6672, 30565, 135668, 587426, 2493056, 10407393, 42848800, 174348417, 702245128, 2803634370, 11106624804, 43697519013, 170871040752, 664492915061, 2571316718500, 9905232077842, 38000679280352, 145240335213857, 553203971301184, 2100403987129441, 7951405959127848

Offset: 1

Christian G. Bower

N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (12, -54, 115, -132, 108, -59, 24, -6, 1).

Cf. A000290, A030267, A279282.

CoefficientList[Series[((1-x)^3(1+x)(1-2x+4x^2-x^3))/((1-4x+2x^2-x^3)^3),{x,0,30}],x] (* or *) LinearRecurrence[{12,-54,115,-132,108,-59,24,-6,1},{1,8,50,276,1397,6672,30565,135668,587426},30] (* Harvey P. Dale, Mar 14 2016 *)

Vec(((1-x)^3*(1+x)*(1-2*x+4*x^2-x^3))/((1-4*x+2*x^2-x^3)^3)+O(x^66)) \\ Joerg Arndt, Apr 21 2013

G.f.: ((1-x)^3*(1+x)*(1-2*x+4*x^2-x^3))/((1-4*x+2*x^2-x^3)^3).

a(n) = 12*a(n-1)-54*a(n-2)+115*a(n-3)-132*a(n-4)+108*a(n-5)-59*a(n-6)+24*a(n-7)-6*a(n-8)+a(n-9). - Wesley Ivan Hurt, Apr 23 2021

A038738 Path-counting array T(i,j) obtained from array t in A038792 by T(i,j)=t(2i+1,j).

Original entry on oeis.org

1, 1, 3, 1, 5, 8, 1, 7, 17, 21, 1, 9, 30, 50, 55, 1, 11, 47, 103, 138, 144, 1, 13, 68, 188, 314, 370, 377, 1, 15, 93, 313, 643, 895, 979, 987, 1, 17, 122, 486, 1201, 1993, 2455, 2575, 2584, 1, 19, 155, 715, 2080, 4082, 5798, 6590, 6755

Offset: 1

Clark Kimberling, May 02 2000

T(n,n)=A001906(n) for n >= 0 (even-indexed Fibonacci numbers).

Row sums: A030267.

			Rows: {1}; {1,3}; {1,5,8}; {1,7,17,21}; ...

A270863 Self-composition of the Fibonacci sequence.

Original entry on oeis.org

0, 1, 2, 6, 17, 50, 147, 434, 1282, 3789, 11200, 33109, 97878, 289354, 855413, 2528850, 7476023, 22101326, 65338038, 193158521, 571033600, 1688143881, 4990651642, 14753839486, 43616704857, 128943855250, 381196100507, 1126928202714, 3331532438042, 9848993360069

Offset: 0

Oboifeng Dira, Mar 24 2016

This sequence has the same relation to the Fibonacci numbers A000045 as A030267 has to the natural numbers A000027.
From Oboifeng Dira, Jun 28 2020: (Start)
This sequence can be generated from a family of composition pairs of generating functions g(f(x)), where k is an integer and where
f(x) = x/(1-k*x-x^2) and g(x) = (x+(k-1)*x^2)/(1-(3-2*k)*x-(3*k-k^2-1)*x^2).
Some cases of k values are:
k=-5, f(x) g.f. 0,A052918(-1)^n and g(x) g.f. 0,A081571
k=-4, f(x) g.f. A001076(-1)^(n+1) and g(x) g.f. 0,A081570
k=-3, f(x) g.f. A006190(-1)^(n+1) and g(x) g.f. 0,A081569
k=-2, f(x) g.f. A215936(n+2) and g(x) g.f. 0,A081568
k=-1, f(x) g.f. A039834(n+2) and g(x) g.f. 0,A081567
k=0,  f(x) g.f. A000035 and g(x) g.f. 0,A001519(n+1)
k=1,  f(x) g.f. A000045 and g(x) g.f. A000045
k=2,  f(x) g.f. A000129 and g(x) g.f. 0,A039834(n+1)
k=3,  f(x) g.f. A006190 and g(x) g.f. 0,A001519(-1)^n
k=4,  f(x) g.f. A001076 and g(x) g.f. 0,A093129(-1)^n
k=5,  f(x) g.f. 0,A052918 and g(x) g.f. 0,A192240(-1)^n
k=6,  f(x) g.f. A005668 and g(x)=(x+5*x^2)/(1+9*x+19*x^2)
k=7,  f(x) g.f. 0,A054413 and g(x)=(x+6*x^2)/(1+11*x+29*x^2).
(End)

			a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50.

Colin Barker, Table of n, a(n) for n = 0..1000
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Oboifeng Dira, Family of composition pairs g(f(x)) generating A270683
Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).

Cf. A000027, A000045, A001622, A030267, A000035, A001519, A000129, A039834.

I:=[0, 1, 2, 6]; [m le 4 select I[m] else 3*Self(m-1)+Self(m-2)-3*Self(m-3)-Self(m-4): m in [1..30]]; // Marius A. Burtea, Aug 03 2019

f:= x-> x/(1-x-x^2):
a:= n-> coeff(series(f(f(x)), x, n+1), x, n):
seq(a(n), n=0..30);

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-3,1,3]^(n-1)*[1;2;6;17])[1,1] \\ Charles R Greathouse IV, Mar 24 2016

concat(0, Vec(x*(1-x-x^2)/(1-3*x-x^2+3*x^3+x^4) + O(x^40))) \\ Colin Barker, Mar 24 2016

a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=2, a(3)=6.
G.f.: x*(1-x-x^2) / (1-3*x-x^2+3*x^3+x^4). - Colin Barker, Mar 24 2016
G.f.: B(B(x)) where B(x) is the g.f. of A000045. - Joerg Arndt, Mar 25 2016
a(n) = (phi*((phi^2 + 5^(1/4)*sqrt(3*phi))^n - (phi^2 - 5^(1/4)*sqrt(3*phi))^n) + (psi^2 + 5^(1/4)*sqrt(3*psi))^n - (psi^2 - 5^(1/4)*sqrt(3*psi))^n)/(2^n * 5^(3/4) * sqrt(3*phi)), where phi = (sqrt(5) + 1)/2 is the golden ratio, and psi = 1/phi = (sqrt(5) - 1)/2. - Vladimir Reshetnikov, Aug 01 2019
0 = a(n)*(a(n) +6*a(n+1) -a(n+2)) +a(n+1)*(8*a(n+1) -9*a(n+2) +a(n+3)) +a(n+2)*(-8*a(n+2) +6*a(n+3)) +a(n+3)*(-a(n+3)) if n>=0. - Michael Somos, Feb 05 2022

A121462 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having pyramid weight k (1 <= k <= n).

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 4, 8, 0, 1, 5, 12, 16, 0, 1, 6, 18, 32, 32, 0, 1, 7, 25, 56, 80, 64, 0, 1, 8, 33, 88, 160, 192, 128, 0, 1, 9, 42, 129, 280, 432, 448, 256, 0, 1, 10, 52, 180, 450, 832, 1120, 1024, 512, 0, 1, 11, 63, 242, 681, 1452, 2352, 2816, 2304, 1024, 0, 1, 12, 75, 316

Offset: 1

Emeric Deutsch, Jul 31 2006

A pyramid in a Dyck word (path) is a factor of the form U^h D^h, where U=(1,1), D=(1,-1) and h is the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.
Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,n)=2^(n-1). Sum_{k=1..n} k*T(n,k) = A030267(n).
Mirror image of triangle in A153342. - Philippe Deléham, Dec 31 2008
Essentially triangle given by (0,1/2,1/2,0,0,0,0,0,0,0,...) DELTA (2,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 30 2011
A121462 is jointly generated with A208341 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = x*u(n-1,x) + x*v(n-1) and v(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 11 2012

			T(4,3)=4 because we have (UD)U(UD)(UD)D, U(UD)(UD)(UD)D, U(UD)(UUDD)D and U(UUDD)(UD)D, where U=(1,1) and D=(1,-1) (the maximal pyramids are shown between parentheses).
Triangle starts:
  1;
  0,  2;
  0,  1,  4;
  0,  1,  4,  8;
  0,  1,  5, 12, 16;
  0,  1,  6, 18, 32, 32;

Elena Barcucci, Alberto del Lungo, S. Fezzi and Renzo Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
Alain Denise and Rodica Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
Emeric Deutsch and Helmut Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.

Cf. A001519, A030267, A091866.

T:=proc(n,k) if n=1 and k=1 then 1 elif k=1 then 0 elif k<=n then sum(binomial(k-1,j)*binomial(n-k-1+j,j-1),j=0..k-1) else 0 fi end: for n from 1 to 13 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1) v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A121462 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208341 *)
(* Clark Kimberling, Mar 11 2012 *)

T(n,k) = Sum_{j=0..k-1} binomial(k-1,j)*binomial(n-k-1+j,j-1) for 2 <= k <= n; T(1,1)=1; T(n,1)=0 for n >= 2.
G.f.: G = G(t,z) = tz(1-z)/(1-2tz-z+tz^2).
T(n+1,k+1) = A062110(n,k)*2^(2*k-n). - Philippe Deléham, Aug 01 2006

A326346 Total number of partitions in the partitions of compositions of n.

Original entry on oeis.org

0, 1, 4, 14, 47, 151, 474, 1457, 4414, 13210, 39155, 115120, 336183, 976070, 2819785, 8110657, 23239662, 66362960, 188930728, 536407146, 1519205230, 4293061640, 12106883585, 34079016842, 95762829405, 268670620736, 752676269695, 2105751165046, 5883798478398

Offset: 0

Alois P. Heinz, Sep 11 2019

nonn

			a(3) = 14 = 1+1+1+2+2+2+2+3 counts the partitions in 3, 21, 111, 2|1, 11|1, 1|2, 1|11, 1|1|1.

Alois P. Heinz, Table of n, a(n) for n = 0..2313

Cf. A000041, A030267, A055887, A060642, A304969, A325915, A327548.

b:= proc(n) option remember; `if`(n=0, [1, 0], (p-> p+
      [0, p[1]])(add(combinat[numbpart](j)*b(n-j), j=1..n)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..32);

b[n_] := b[n] = If[n==0, {1, 0}, Function[p, p + {0, p[[1]]}][Sum[ PartitionsP[j] b[n-j], {j, 1, n}]]];
a[n_] := b[n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A060642(n,k).

a(n) ~ c * d^n * n, where d = A246828 = 2.69832910647421123126399866618837633... and c = 0.171490233695958246364725709205670983251448838158816... - Vaclav Kotesovec, Sep 14 2019

A188137 Riordan array (1, x(1-x)/(1-3x+x^2)).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 13, 14, 6, 1, 34, 46, 27, 8, 1, 89, 145, 107, 44, 10, 1, 233, 444, 393, 204, 65, 12, 1, 610, 1331, 1371, 854, 345, 90, 14, 1, 1597, 3926, 4607, 3336, 1620, 538, 119, 16, 1, 4181, 11434, 15045, 12390, 6997, 2799, 791, 152, 18, 1

Offset: 1

Vladimir Kruchinin, Mar 21 2011

The column of index 0 contains a 1 followed by zeros and is not reproduced in this triangle.
The second argument of the array definition is A(x) = A000045(x/(1-x)) = A001519(x)-1.
Triangle T(n,k), 1 <= k <= n, given by (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 26 2012

			Triangle begins:
   1;
   2,   1;
   5,   4,   1;
  13,  14,   6,  1;
  34,  46,  27,  8,  1;
  89, 145, 107, 44, 10, 1;
From _Philippe Deléham_, Jan 26 2012: (Start)
Triangle (0,2,1/2,1/2,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,...) begins:
  1;
  0,  1;
  0,  2,   1;
  0,  5,   4,   1;
  0, 13,  14,   6,  1;
  0, 34,  46,  27,  8,  1;
  0, 89, 145, 107, 44, 10, 1; (End)

Alois P. Heinz, Rows n = 1..141, flattened
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A001519 (column 1), A030267 (column 2).

A188137 := proc(n,m) add( binomial(n-1,k-1) *add(binomial(i,k-m-i) *binomial(m+i-1,m-1),i=ceil((k-m)/2)..k-m),k=m..n) ; end proc:
seq(seq(A188137(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Mar 30 2011

t[n_, m_] := Sum[ Binomial[n - 1, k - 1]*Sum[ Binomial[i, k - m - i]*Binomial[m + i - 1, m - 1], {i, Ceiling[(k - m)/2], k - m}], {k, m, n}]; Table[t[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013, translated from Maxima *)

T(n,m):=sum(binomial(n-1,k-1) *sum(binomial(i,k-m-i) *binomial(m+i-1,m-1), i,ceiling((k-m)/2),k-m), k,m,n);

T(n,m) = Sum_{k=m..n} binomial(n-1,k-1) * Sum_{i=ceiling((k-m)/2)..k-m} binomial(i,k-m-i)*binomial(m+i-1,m-1), 0

T(n,m) = Sum_{i=1..n-m+1} A001519(i)*T(n-i,m-1).

T(n,1) = A001519(n).

Sum_{m=1..n} T(n,m) = A007052(n-1).

G.f.: (1-3x+x^2)/(1-(3+y)*x + (1+y)*x^2). - Philippe Deléham, Jan 26 2012

A119749 Number of compositions of n into odd blocks with one element in each block distinguished.

Original entry on oeis.org

1, 1, 4, 7, 15, 32, 65, 137, 284, 591, 1231, 2560, 5329, 11089, 23076, 48023, 99935, 207968, 432785, 900633, 1874236, 3900319, 8116639, 16890880, 35150241, 73148321, 152223044, 316779047, 659223215, 1371856032, 2854858465

Offset: 1

Louis Shapiro, Jul 30 2006

The sequence is the INVERT transform of the aerated odd integers. - Gary W. Adamson, Feb 02 2014

Number of compositions of n into odd parts where there is 1 sort of part 1, 3 sorts of part 3, 5 sorts of part 5, ... , 2*k-1 sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014

			a(3) = 4 since Abc, aBc, abC come from one block of size 3 and A/B/C comes from having three blocks. The capital letters are the distinguished elements.

R. X. F. Chen and L. W. Shapiro, On Sequences G(n) satisfying G(n)=(d+2)*G(n-1)-G(n-2), J. Int. Seq. 10 (2007) #07.8.1, Theorem 16.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (6).
Y.-h. Guo, n-Color Odd Self-Inverse Compositions, J. Int. Seq. 17 (2014) # 14.10.5, eq. (2).
B. Hopkins, Spotted tilings and n-color compositions, INTEGERS 12B (2012/2013), #A6.
Index entries for linear recurrences with constant coefficients, signature (1,2,1,-1).

Cf. A105309, A052530, A000045, A030267. Row sums of A292835.

Rest@ CoefficientList[ Series[x(1 + x^2)/(x^4 - x^3 - 2x^2 - x + 1), {x, 0, 50}], x] (* Robert G. Wilson v *)

G.f.: (x+x^3)/(x^4 - x^3 -2x^2 -x +1).
a(n) = A092886(n)+A092886(n-2). - R. J. Mathar, Mar 08 2018
Sum_{k=0..n} a(k) = (3*a(n) + 2*a(n-1) - a(n-3))/2 - 1. - Xilin Wang and Greg Dresden, Aug 27 2020

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cp's OEIS Frontend

A001870 Expansion of (1-x)/(1 - 3*x + x^2)^2.

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A038731 Number of columns in all directed column-convex polyominoes of area n+1.

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A153294 G.f.: A(x) = F(x*F(x)^2) = F(F(x)-1) where F(x) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan).

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A030279 COMPOSE squares with squares.

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A038738 Path-counting array T(i,j) obtained from array t in A038792 by T(i,j)=t(2i+1,j).

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A270863 Self-composition of the Fibonacci sequence.

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A121462 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having pyramid weight k (1 <= k <= n).

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A153294 G.f.: A(x) = F(xF(x)^2) = F(F(x)-1) where F(x) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan).

A188137 Riordan array (1, x(1-x)/(1-3x+x^2)).