A038112 -id:A038112 - cp's OEIS frontend

A037027 Skew Fibonacci-Pascal triangle read by rows.

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 10, 9, 4, 1, 8, 20, 22, 14, 5, 1, 13, 38, 51, 40, 20, 6, 1, 21, 71, 111, 105, 65, 27, 7, 1, 34, 130, 233, 256, 190, 98, 35, 8, 1, 55, 235, 474, 594, 511, 315, 140, 44, 9, 1, 89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1, 144, 744, 1836

Offset: 0

Floor van Lamoen, Jan 01 1999

T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (0,1), (1,0), (2,0). - Joerg Arndt, Jun 30 2011
T(n,k) is the number of lattice paths of length n, starting from the origin and ending at (n,k), using horizontal steps H=(1,0), up steps U=(1,1) and down steps D=(1,-1), never containing UUU, DD, HD. For instance, for n=4 and k=2, we have the paths; HHUU, HUHU, HUUH, UHHU, UHUH, UUHH, UUDU, UDUU, UUUD. - Emanuele Munarini, Mar 15 2011
Row sums form Pell numbers A000129, T(n,0) forms Fibonacci numbers A000045, T(n,1) forms A001629. T(n+k,n-k) is polynomial sequence of degree k.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, etc.).
As a Riordan array, this is (1/(1-x-x^2),x/(1-x-x^2)). An interesting factorization is (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x)) [abs(A049310) times A007318]. Diagonal sums are the Jacobsthal numbers A001045(n+1). - Paul Barry, Jul 28 2005
T(n,k) = T'(n+1,k+1), T' given by [0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2005
Equals A049310 * A007318 as infinite lower triangular matrices. - Gary W. Adamson, Oct 28 2007
This triangle may also be obtained from the coefficients of the Morgan-Voyce polynomials defined by: Mv(x, n) = (x + 1)*Mv(x, n - 1) + Mv(x, n - 2). - Roger L. Bagula, Apr 09 2008
Row sums are A000129. - Roger L. Bagula, Apr 09 2008
Absolute value of coefficients of the characteristic polynomial of tridiagonal matrices with 1's along the main diagonal, and i's along the superdiagonal and the subdiagonal (where i=sqrt(-1), see Mathematica program). - John M. Campbell, Aug 23 2011
A037027 is jointly generated with A122075 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x). See the Mathematica section at A122075. - Clark Kimberling, Mar 05 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
Row n, for n>=0, shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^n which is the denominator of the n-th convergent of the continued fraction [x+1, x+1, x+1, ...]; see A230000. - Clark Kimberling, Nov 13 2013
T(n,k) is the number of ternary words of length n having k letters 2 and avoiding a runs of odd length for the letter 0. - Milan Janjic, Jan 14 2017
Let T(m, n, k) be an m-bonacci Pascal's triangle, where T(m, n, 0) gives the values of F(m, n), the n-th m-bonacci number, and T(m, n, k) gives the values for the k-th convolution of F(m, n). Then the classic Pascal triangle is T(1, n, k) and this sequence is T(2, n, k). T(m, n, k) is the number of compositions of n using only the positive integers 1, 1' and 2 through m, with the part 1' used exactly k times. G.f. for k-th column of T(m, n, k): x/(1 - x - x^2 - ... - x^m)^k. The row sum for T(m, n, k) is the number of compositions of n using only the positive integers 1, 1' and 2 through m. G.f. for row sum of T(m, n, k): 1/(1 - 2x - x^2 - ... - x^m). - Gregory L. Simay, Jul 24 2021

			Ratio of row polynomials R(3)/R(2) = (3 + 5*x + 3*x^2 + x^3)/(2 + 2*x + x^2) = [1+x; 1+x, 1+x].
Triangle begins:
                                 1;
                              1,    1;
                           2,    2,    1;
                        3,    5,    3,    1;
                     5,   10,    9,    4,    1;
                  8,   20,   22,   14,    5,    1;
              13,   38,   51,   40,   20,    6,    1;
           21,   71,  111,  105,   65,   27,    7,    1;
        34,  130,  233,  256,  190,   98,   35,    8,    1;
     55,  235,  474,  594,  511,  315,  140,   44,    9,    1;
  89,  420,  942, 1324, 1295,  924,  490,  192,   54,   10,    1;

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Harlan J. Brothers, Pascal's Prism: Supplementary Material
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
T. Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials.

A038112(n) = T(2n, n). A038137 is reflected version. Maximal row entries: A038149.
Diagonal differences are in A055830. Vertical sums are in A091186.
Cf. A007318, A049310, A000129, A155161, A122542, A059283, A228196, A228576.
Some other Fibonacci-Pascal triangles: A027926, A036355, A074829, A105809, A109906, A111006, A114197, A162741, A228074.

a037027 n k = a037027_tabl !! n !! k
a037027_row n = a037027_tabl !! n
a037027_tabl = [1] : [1,1] : f [1] [1,1] where
   f xs ys = ys' : f ys ys' where
     ys' = zipWith3 (\u v w -> u + v + w) (ys ++ [0]) (xs ++ [0,0]) ([0] ++ ys)
-- Reinhard Zumkeller, Jul 07 2012

T := (n,k) -> `if`(n=0,1,binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -4)): seq(seq(simplify(T(n,k)), k=0..n), n=0..10); # Peter Luschny, Apr 25 2016
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022

Mv[x, -1] = 0; Mv[x, 0] = 1; Mv[x, 1] = 1 + x; Mv[x_, n_] := Mv[x, n] = ExpandAll[(x + 1)*Mv[x, n - 1] + Mv[x, n - 2]]; Table[ CoefficientList[ Mv[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Apr 09 2008 *)
Abs[Flatten[Table[CoefficientList[CharacteristicPolynomial[Array[KroneckerDelta[#1,#2]+KroneckerDelta[#1,#2+1]*I+KroneckerDelta[#1,#2-1]*I&,{n,n}],x],x],{n,1,20}]]] (* John M. Campbell, Aug 23 2011 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k-n)/2, (k-n+1)/2, -n, -4];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 16 2019, after Peter Luschny *)

{T(n, k) = if( k<0 || k>n, 0, if( n==0 && k==0, 1, T(n-1, k) + T(n-1, k-1) + T(n-2, k)))}; /* Michael Somos, Sep 29 2003 */

PARI
```
T(n,k)=if(nPaul D. Hanna, Feb 27 2004
```

T(n, m) = T'(n-1, m) + T'(n-2, m) + T'(n-1, m-1), where T'(n, m) = T(n, m) for n >= 0 and 0< = m <= n and T'(n, m) = 0 otherwise.
G.f.: 1/(1 - y - y*z - y^2).
G.f. for k-th column: x/(1-x-x^2)^k.
T(n, m) = Sum_{k=0..n-m} binomial(m+k, m)*binomial(k, n-k-m), n >= m >= 0, otherwise 0. - Wolfdieter Lang, Jun 17 2002
T(n, m) = ((n-m+1)*T(n, m-1) + 2*(n+m)*T(n-1, m-1))/(5*m), n >= m >= 1; T(n, 0)= A000045(n+1); T(n, m)= 0 if n < m. - Wolfdieter Lang, Apr 12 2000
Chebyshev coefficient triangle (abs(A049310)) times Pascal's triangle (A007318) as product of lower triangular matrices. T(n, k) = Sum_{j=0..n} binomial((n+j)/2, j)*(1+(-1)^(n+j))*binomial(j, k)/2. - Paul Barry, Dec 22 2004
Let R(n) = n-th row polynomial in x, with R(0)=1, then R(n+1)/R(n) equals the continued fraction [1+x;1+x, ...(1+x) occurring (n+1) times ..., 1+x] for n >= 0. - Paul D. Hanna, Feb 27 2004
T(n,k) = Sum_{j=0..n} binomial(n-j,j)*binomial(n-2*j,k); in Egorychev notation, T(n,k) = res_w(1-w-w^2)^(-k-1)*w^(-n+k+1). - Paul Barry, Sep 13 2006
Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. - Philippe Deléham, Nov 29 2009
T((m+1)*n+r-1, m*n+r-1)*r/(m*n+r) = Sum_{k=1..n} k/n*T((m+1)*n-k-1, m*n-1)*(r+k,r), n >= m > 1.
T(n-1,m-1) = (m/n)*Sum_{k=1..n-m+1} k*A000045(k)*T(n-k-1,m-2), n >= m > 1. - Vladimir Kruchinin, Mar 17 2011
T(n,k) = binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -4) for n >= 1. - Peter Luschny, Apr 25 2016

Examples from Paul D. Hanna, Feb 27 2004

A001002 Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.

Original entry on oeis.org

1, 1, 3, 10, 38, 154, 654, 2871, 12925, 59345, 276835, 1308320, 6250832, 30142360, 146510216, 717061938, 3530808798, 17478955570, 86941210950, 434299921440, 2177832612120, 10959042823020, 55322023332420, 280080119609550, 1421744205767418, 7234759677699954

Offset: 0

N. J. A. Sloane

a(n+1) is number of (2,3)-rooted trees on n nodes.
This sequence appears to be a transform of the Fibonacci numbers A000045. This sequence is to the Fibonacci numbers as the Catalan numbers A000108 is to the all ones sequence. See link to Mathematica program. - Mats Granvik, Dec 30 2017
a(n) is the number of parking functions of size n avoiding the patterns 231, 312, and 321. - Lara Pudwell, Apr 10 2023

			a(3)=10 because a convex pentagon can be dissected in 5 ways into triangles (draw 2 diagonals from any of the 5 vertices) and in 5 ways into a triangle and a quadrilateral (draw any of the 5 diagonals).

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 211 (3.2.73-74)
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1372 (first 101 terms from T. D. Noe)
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Paul Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3, example 7.
D. Birmajer, J. B. Gil, and M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
I. M. H. Etherington, Non-associate powers and a functional equation, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.
I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. [Annotated scanned copy].
A. Erdelyi and I. M. H. Etherington, Some problems of non-associative combinations (II), Edinburgh Math. Notes, 32 (1940), pp. vii-xiv.
Mats Granvik, Mathematica program to compute the sequence as a transform of the Fibonacci numbers.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 395
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.
A. Schuetz and G. Whieldon, Polygonal Dissections and Reversions of Series, arXiv preprint arXiv:1401.7194 [math.CO], 2014.
Index entries for reversions of series

n*a(n) = A038112(n-1), n > 0.

Cf. A104978, A181997, A181998, A209441, A209442, A213684 (log).

List([0..25], n->Sum([0..Int(n/2)],k->Binomial(2*n-k,n+k)*Binomial(n+k,k)/(n+1))); # Muniru A Asiru, Mar 30 2018

a:= proc(n) option remember; `if`(n<2, 1, (n*(22*n-11)*
      a(n-1) + (9*n-6)*(3*n-4)*a(n-2))/(5*n*(n+1)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 21 2021

Rest[CoefficientList[InverseSeries[Series[y - y^2 - y^3, {y, 0, 30}], x], x]]
a[n_] := CatalanNumber[n]*Hypergeometric2F1[1/2-n/2, -n/2, -2n, -4]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 20 2015, after Peter Luschny *)
a[n_] := a[n] = If[n == 0, 1, Sum[a[i] a[n - 1 - i], {i, 0, n - 1}] + Sum[a[i] a[j] a[n - 2 - i - j], {i, 0, n - 2}, {j, 0, n - 2 - i}]];
Table[a[n], {n, 0, 30}] (* Li Han, Jan 02 2021 *)

T(n,k):=if n<0 or k<0 then 0 else if nVladimir Kruchinin, Oct 03 2014 */

a(n)=if(n<0,0,polcoeff(serreverse(x-x^2-x^3+x^2*O(x^n)),n+1))

a(n)=if(n<0,0,sum(k=0,n\2,(2*n-k)!/k!/(n-2*k)!)/(n+1)!)

a(n)=sum(k=0,n\2,binomial(2*n-k,n+k)*binomial(n+k,k))/(n+1) \\ Hanna

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=1+(1/x)*sum(m=1, n+1, Dx(m-1, (x^2+x^3 +x^2*O(x^n))^m/m!)); polcoeff(A, n)}  \\ Paul D. Hanna, Jun 22 2012

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=exp(sum(m=1, n+1, Dx(m-1, (x^2+x^3 +x^2*O(x^n))^m/x/m!)+x*O(x^n))); polcoeff(A, n)}  \\ Paul D. Hanna, Jun 22 2012

A001002 = lambda n: catalan_number(n)*hypergeometric([1/2-n/2, -n/2], [-2*n], -4) if n>0 else 1
[A001002(n).n(100).round() for n in range(24)] # Peter Luschny, Oct 03 2014

G.f. (offset 1) is series reversion of x - x^2 - x^3.
a(n) = (1/(n+1))*Sum_{k=ceiling(n/2)..n} binomial(n+k, k)*binomial(k, n-k). - Len Smiley
D-finite with recurrence 5*n*(n+1) * a(n) = 11*n*(2*n-1) * a(n-1) + 3*(3*n-2)*(3*n-4) * a(n-2). - Len Smiley
G.f.: (4*sin(asin((27*x+11)/16)/3)-1)/(3*x). - Paul Barry, Feb 02 2005
G.f. satisfies: A(x) = 1 + x*A(x)^2 + x^2*A(x)^3. - Paul D. Hanna, Jun 22 2012
Antidiagonal sums of triangle A104978 which has g.f. F(x,y) that satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D. Hanna, Mar 30 2005
a(n) = Sum_{k=0..floor(n/2)} C(2*n-k, n+k)*C(n+k, k)/(n+1). - Paul D. Hanna, Mar 30 2005
G.f. satisfies: x = Sum_{n>=1} 1/(1+x*A(x))^(2*n) * Product_{k=1..n} (1 - 1/(1+x*A(x))^k). - Paul D. Hanna, Apr 05 2012
G.f.: 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) (x^2+x^3)^n/n!. - Paul D. Hanna, Jun 22 2012
G.f.: exp( Sum_{n>=1} d^(n-1)/dx^(n-1) ((x^2+x^3)^n/x)/n! ). - Paul D. Hanna, Jun 22 2012
Logarithmic derivative yields A213684. - Paul D. Hanna, Jun 22 2012
a(n) ~ 3^(3*n+3/2) / (2 * sqrt(2*Pi) * 5^(n+1/2) * n^(3/2)). - Vaclav Kotesovec, Mar 09 2014
a(n) = Catalan(n)*hypergeom([1/2-n/2, -n/2], [-2*n], -4) for n>0. - Peter Luschny, Oct 03 2014
a(n) = [x^n] 1/(1 - x - x^2)^(n+1)/(n + 1). - Ilya Gutkovskiy, Mar 29 2018
a(n) = -Sum_{i=1..n} A217596(i) * a(n-i) for n>0. - Muhammed Sefa Saydam, Jan 27 2025
a(n) = -Sum_{i=1..n+2} A217596(i) * A217596(n-i+2) for n >= 0. - Muhammed Sefa Saydam, Jul 24 2025

Revised by Emeric Deutsch and Len Smiley, Jun 05 2005

A111006 Another version of Fibonacci-Pascal triangle A037027.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144

Offset: 0

Philippe Deléham, Oct 02 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.
Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, ...).
Triangle read by rows: T(n,n-k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)). - Philippe Deléham, Feb 17 2014
Diagonal sums are A013979(n). - Philippe Deléham, Feb 17 2014
T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and 1 X 2 tiles. - Emeric Deutsch, Aug 14 2014

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 1, 5,  5;
  0, 0, 0, 3, 10,  8;
  0, 0, 0, 1,  9, 20, 13;
  0, 0, 0, 0,  4, 22, 38,  21;
  0, 0, 0, 0,  1, 14, 51,  71,  34;
  0, 0, 0, 0,  0,  5, 40, 111, 130,  55;
  0, 0, 0, 0,  0,  1, 20, 105, 233, 235,  89;
  0, 0, 0, 0,  0,  0,  6,  65, 256, 474, 420, 144;

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000045, A000129, A001045, A037027, A038112, A038149, A084938, A128100 (reversed version).

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A114197, A162741, A228074.

a111006 n k = a111006_tabl !! n !! k
a111006_row n = a111006_tabl !! n
a111006_tabl =  map fst $ iterate (\(us, vs) ->
   (vs, zipWith (+) (zipWith (+) ([0] ++ us ++ [0]) ([0,0] ++ us))
                    ([0] ++ vs))) ([1], [0,1])
-- Reinhard Zumkeller, Aug 15 2013

T(0, 0) = 1, T(n, k) = 0 for k < 0 or for n < k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).
T(n, k) = A037027(k, n-k). T(n, n) = A000045(n+1). T(3n, 2n) = (n+1)*A001002(n+1) = A038112(n).
G.f.: 1/(1-yx(1-x)-x^2*y^2). - Paul Barry, Oct 04 2005
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 02 2006
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 respectively. - Philippe Deléham, Feb 17 2014

A215128 G.f.: Sum_{n>=0} d^n/dx^n (x + x^2)^(2*n) / n!.

Original entry on oeis.org

1, 2, 12, 64, 370, 2184, 13132, 79944, 491238, 3040400, 18926336, 118369368, 743199184, 4681668488, 29574616440, 187281906512, 1188494457492, 7556371963488, 48123031011036, 306929964849200, 1960230225450420, 12534313062502440, 80236414444623240

Offset: 0

Paul D. Hanna, Aug 04 2012

nonn

Compare to: Sum_{n>=0} d^n/dx^n x^(2*n)/n! = 1/sqrt(1-4*x).

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 64*x^3 + 370*x^4 + 2184*x^5 + 13132*x^6 +...
such that, by definition:
A(x) = 1 + d/dx (x+x^2)^2 + d^2/dx^2 (x+x^2)^4/2! + d^3/dx^3 (x+x^2)^6/3! + d^4/dx^4 (x+x^2)^8/4! + d^5/dx^5 (x+x^2)^10/5! +...

Cf. A214372, A215125, A215129, A038112.

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

a(n) = (n+1)*A214372(n+1), where G(x) = x + (G(x) + G(x)^2)^2 is the g.f. of A214372.

A383479 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(3,0),(0,1).

Original entry on oeis.org

1, 2, 6, 24, 100, 420, 1792, 7752, 33858, 148940, 658944, 2929056, 13070876, 58521344, 262754040, 1182619280, 5334172518, 24104916504, 109111142376, 494630028200, 2245300152480, 10204575481320, 46429481139000, 211460450151600, 963971663881200, 4398118872144192

Offset: 0

Seiichi Manyama, Apr 28 2025

nonn

Robert Israel, Table of n, a(n) for n = 0..1492

Cf. A038112, A383480.

Cf. A049140, A144401, A370624.

f:= proc(x,y) option remember;
     local t;
     t:= 0;
     if x >= 1 then t:= t + procname(x-1,y) fi;
     if x >= 3 then t:= t + procname(x-3,y) fi;
     if y >= 1 then t:= t + procname(x,y-1) fi;
     t
end proc:
f(0,0):= 1:
seq(f(n,n),n=0..25); # Robert Israel, May 28 2025

a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(2*n-2*k, n-3*k));

a(n) = [x^n] 1/(1 - x - x^3)^(n+1).
a(n) = (n+1) * A049140(n+1).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(2*n-2*k,n-3*k).

A383480 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(4,0),(0,1).

Original entry on oeis.org

1, 2, 6, 20, 75, 294, 1176, 4752, 19350, 79310, 326898, 1353768, 5628441, 23478700, 98217840, 411879264, 1730924700, 7287941340, 30736775190, 129825892000, 549096132585, 2325216522420, 9857299586700, 41830206233400, 177673556967075, 755307883986084, 3213402383779812

Offset: 0

Seiichi Manyama, Apr 28 2025

nonn

Robert Israel, Table of n, a(n) for n = 0..1564

Cf. A038112, A383479.

Cf. A063021, A383481.

f:= proc(x,y) option remember;
     local t;
     t:= 0;
     if x >= 1 then t:= t + procname(x-1,y) fi;
     if x >= 4 then t:= t + procname(x-4,y) fi;
     if y >= 1 then t:= t + procname(x,y-1) fi;
     t
end proc:
f(0,0):= 1:
seq(f(n,n),n=0..26); # Robert Israel, May 28 2025

a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(2*n-3*k, n-4*k));

a(n) = [x^n] 1/(1 - x - x^4)^(n+1).
a(n) = (n+1) * A063021(n+1).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+k,k) * binomial(2*n-3*k,n-4*k).

A231373 G.f. A(x) satisfies: A(x-x^2-x^3) = 1/sqrt(1-2x-3x^2), which is the g.f. the central trinomial coefficients (A002426).

Original entry on oeis.org

1, 1, 4, 16, 71, 327, 1550, 7490, 36720, 182028, 910330, 4585318, 23233722, 118315318, 605088690, 3105994302, 15994906965, 82602799485, 427662046960, 2219130114108, 11538302709769, 60102637378353, 313591732265662, 1638671208390738, 8574718477933404, 44926247350136232

Offset: 0

Paul D. Hanna, Nov 08 2013

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 71*x^4 + 327*x^5 + 1550*x^6 +...
where A(x-x^2-x^3)^2 = 1/(1-2*x-3*x^2):
A(x-x^2-x^3) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 141*x^6 + 393*x^7 + 1107*x^8 +...+ A002426(n)*x^n +...
The square of the g.f. begins (cf. A038112):
A(x)^2 = 1 + 2*x + 9*x^2 + 40*x^3 + 190*x^4 + 924*x^5 + 4578*x^6 +...
such that A(x)^2 = d/dx x*G(x) where G(x) is the g.f. of A001002:
G(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 154*x^5 + 654*x^6 +...
and satisfies G(x-x^2-x^3) = 1/(1-x-x^2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A002426, A038112, A001002.

CoefficientList[Series[Sqrt[D[InverseSeries[Series[x - x^2 - x^3, {x, 0, 30}], x], x]], {x, 0, 30}], x] (* Vaclav Kotesovec, Mar 31 2014 *)

{a(n)=local(G=serreverse(x-x^2-x^3+x^2*O(x^n)),A);A=sqrt(deriv(G));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ = d^n/dx^n F
{a(n)=local(A2=x); A2=1+sum(m=1, n+1, Dx(m, x^(2*m)*(1+x +x*O(x^n))^m/m!)); polcoeff(sqrt(A2), n)}
for(n=0,30,print1(a(n),", "))

Self-convolution yields A038112.
G.f. A(x) satisfies:
(1) A(x) = sqrt( Sum_{n>=0} d^n/dx^n x^(2*n)*(1+x)^n/n! ).
(2) A(x) = sqrt((1+x)*(5-27*x)*A(x)^6 - 1)/2, from a formula by Mark van Hoeij in A038112.
(3) A(x) = sqrt( d/dx x*G(x) ) where G(x) = Series_Reversion(x-x^2-x^3)/x is the g.f. of A001002.
(4) A(x) = 1/sqrt(1 - 2*x*G(x) - 3*x^2*G(x)^2) where G(x) = Series_Reversion(x-x^2-x^3)/x is the g.f. of A001002.
Sum_{k=0..n} a(k)*a(n-k) = Sum_{k=0..n} C(n+k, k)*C(k, n-k), from a formula by Paul Barry in A038112.
Recurrence: 25*(n-2)*(n-1)*n*a(n) = 110*(n-2)*(n-1)*(2*n-3)*a(n-1) - (n-2)*(214*n^2 - 856*n + 717)*a(n-2) - 33*(2*n-5)*(18*n^2 - 90*n + 113)*a(n-3) - 81*(n-3)*(3*n-11)*(3*n-7)*a(n-4). - Vaclav Kotesovec, Nov 10 2013
a(n) ~ 3^(3/4) * GAMMA(3/4) * (27/5)^n / (2*10^(1/4)*Pi*n^(3/4)). - Vaclav Kotesovec, Dec 29 2013

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A037027 Skew Fibonacci-Pascal triangle read by rows.

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A001002 Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.

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A111006 Another version of Fibonacci-Pascal triangle A037027.

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A215128 G.f.: Sum_{n>=0} d^n/dx^n (x + x^2)^(2*n) / n!.

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A383479 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(3,0),(0,1).

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A383480 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(4,0),(0,1).

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A231373 G.f. A(x) satisfies: A(x-x^2-x^3) = 1/sqrt(1-2x-3x^2), which is the g.f. the central trinomial coefficients (A002426).