A004148 -id:A004148 - cp's OEIS frontend

A307733 a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-1} a(k) * a(n-k-1).

1, 1, 4, 14, 54, 220, 934, 4090, 18344, 83850, 389214, 1829736, 8693962, 41685714, 201442188, 980091814, 4797070022, 23603701828, 116688837886, 579312087802, 2887020896016, 14437318756818, 72424982972862, 364366674463824, 1837954750285458

Offset: 0

Ilya Gutkovskiy, Jul 05 2020

nonn

Cf. A002212, A004148, A006318, A085139, A128720, A143330, A171416, A175934, A245734.

a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
nmax = 24; CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 2 x^3 + x^4])/(2 x), {x, 0, nmax}], x]

G.f. A(x) satisfies: A(x) = (1 - x + x*A(x)^2) / (1 - x - x^2).

G.f.: (1 - x - x^2 - sqrt(1 - 6*x + 3*x^2 + 2*x^3 + x^4)) / (2*x).

A329672 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps UU.

Original entry on oeis.org

1, 2, 4, 9, 20, 46, 107, 252, 599, 1435, 3460, 8389, 20437, 49996, 122758, 302401, 747114, 1850696, 4595370, 11435380, 28513149, 71225270, 178219696, 446637759, 1120946389, 2817089354, 7088656546, 17858286741, 45039810918, 113711798916, 287369435649, 726905294670, 1840328917065

Offset: 0

Valerie Roitner, Nov 26 2019

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). A meander is a path starting at (0,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.

			a(2)=4 since we have 4 meanders of length 2 avoiding UU, namely UH, UD, HU and HH.

Cf. A004148 (shifted by 1) which counts excursions avoiding consecutive UU steps. See also A329673 and A329674 which count meanders avoiding consecutive HH and DD respectively.

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

D-finite with recurrence (n+2)*a(n) +(-3*n-5)*a(n-1) +(-3*n+2)*a(n-2) +(5*n+2)*a(n-3) +(11*n-19)*a(n-4) +(9*n-32)*a(n-5) +2*a(n-6) +2*(-n+6)*a(n-7)=0. - R. J. Mathar, Jan 25 2023

A329674 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps DD.

Original entry on oeis.org

1, 2, 5, 13, 34, 90, 240, 643, 1729, 4662, 12597, 34095, 92406, 250719, 680877, 1850457, 5032296, 13692674, 37274438, 101509476, 276535824, 753574253, 2054064713, 5600176231, 15271331416, 41651397245, 113618996429, 309979833301, 845805408448, 2308108658854, 6299205562846

Offset: 0

Valerie Roitner, Nov 26 2019

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). A meander is a path starting at (0,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.

			a(2)=5 since we have 5 meanders of length 2 avoiding DD, namely UU, UH, UD, HU and HH.

Cf. A004148 (shifted by 1) which counts excursions avoiding consecutive DD steps.

Cf. A329672 and A329673 which count meanders avoiding consecutive UU or HH respectively.

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

D-finite with recurrence (n+1)*a(n) +(-4*n-1)*a(n-1) +(n-2)*a(n-2) +(4*n+1)*a(n-3) +(7*n-23)*a(n-4) +2*(n-2)*a(n-5) +2*(-n+5)*a(n-6)=0. - R. J. Mathar, Jan 25 2023

A364366 An irregular triangle read by rows, the 3rd row-symmetric Fibonaccian triangle: T(n,k) is the Whitney number of level k of the (3,n)-th symmetric Fibonaccian lattice (0 <= n, 0 <= k <= 2*n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 4, 5, 4, 3, 1, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 1, 6, 16, 30, 46, 58, 63, 58, 46, 30, 16, 6, 1, 1, 7, 22, 47, 81, 116, 143, 158, 143, 116, 81, 47, 22, 7, 1

Offset: 0

Robert G. Donnelly and Molly W. Dunkum, Jul 20 2023

For integers m and n (m >= 2, n > 0), let L be the set of n-tuples S=(S(1),...,S(n)) with each S(j) in {(j-1)*m+1,(j-1)*m+2,...,j*m} and such that S has no consecutive integers. Partially order these '(m,n) Fibonaccian strings' comprising L by the rule R <= S iff R(j) >= S(j) for 1 <= j <= n (so, 'lightest' n-tuples are at the top of the Hasse diagram for L). Then L is a self-dual distributive lattice, the '(m,n)-th symmetric Fibonaccian lattice'. When n=1, L is a chain with m elements. Now allow n=0; in this case, regard L to be a singleton set. Let p(n,x) be the rank generating function of L, so p(n,1)=|L|, p(0,x)=1, and p(1,x)=1+x+...+x^(m-1). For n >= 2, the fact that p(n,x) = p(1,x)*p(n-1,x) - x^(m-1)*p(n-2,x) can be deduced from a recurrence of Whitney numbers of symmetric Fibonaccian lattices proved in Proposition 2.1 of [Donnelly, Dunkum, Lišková, and Nance, 2023].
The (m,n)-th symmetric Fibonaccian lattice realizes a p(n,1)-dimensional representation of the special linear Lie algebra sl(m,C). The representation is reducible exactly when m >= 3 and n >= 3. The polynomial p(n,x) is a natural specialization of the character of this representation, where the latter can be identified as a certain skew Schur function. In [Donnelly, Dunkum, Lišková, and Nance, 2023], these representations are uniformly constructed (as an application of [Donnelly and Dunkum, 2022]) and explicit formulas for p(n,x) are given.
In [Donnelly, Dunkum, Lišková, and Nance, 2023], the (m,n)-th symmetric Fibonaccian lattice L is also described using semistandard tableaux of a specific ribbon shape; the irreducible components of the associated sl(m,C)-representation are in one-to-one correspondence with what are called the 'ballot-admissible' (aka Littlewood-Richardson) tableaux. In terms of Fibonaccian strings, an element S = (S(1),...,S(n)) in L is ballot-admissible iff for any integer q between 1 and n (inclusive) and any integer r between 1 and m-1 (inclusive), the following integer quantity is nonnegative: Sum_{k=n+1-q..n}([n+1-k is odd]*([r+(k-1)*m = S(k)] - [r+(k-1)*m+1 = S(k)]) + [n+1-k is even]*([k*m-r = S(k)]-[k*m+1-r = S(k)])), where '[]' denotes the Iverson bracket. Enumerating the ballot-admissible tableaux or Fibonaccian strings in L seems to be an interesting problem when m >= 3; when m=3, the sizes of the sets of ballot-admissible tableaux conjecturally agree with A004148.
In this OEIS entry, we have m=3. Let L be the (3,n)-th symmetric Fibonaccian lattice. When n=0, we have T(0,0) = |L| = 1. When n=1, we have T(1,0) = T(1,1) = T(1,2) = 1 and p(1,x) = 1+x+x^2, since L is a chain with 3 elements. For n >= 2, we have, by definition, p(n,x) = Sum_{k=0..2*n} T(n,k)*x^k. The Whitney number T(n,k) is the number of (3,n) Fibonaccian strings S=(S(1),...,S(n)) whose coordinate sum S(1)+...+S(n) is equal to 3*(n*(n+1)/2)-k. This irregular triangle of T(n,k)'s is obtained from the regular triangles in entries A079487 and A123245 by removing all even length rows (with the exception of the row of length two, all such even length rows are 'asymmetric').
For m=4, see A364367. For m=5, see A364368. When m=2, the (2,n)-th symmetric Fibonaccian lattice is a chain with n+1 elements and rank generating function 1+x+...+x^(n-1)+x^n. Therefore, the 2nd row-symmetric Fibonaccian triangle is a regular triangle of 1's. The 1st row-symmetric Fibonaccian 'triangle' is regarded to be the signed sequence 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, ... (A010892). 'Gibonaccian' versions of such triangles are considered in [Donnelly, Dunkum, Huber, and Knupp, 2021].

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) starts as follows:
  1;
  1,   1,   1;
  1,   2,   2,   2,   1;
  1,   3,   4,   5,   4,   3,   1;
  1,   4,   7,  10,  11,  10,   7,   4,   1;
  1,   5,  11,  18,  24,  26,  24,  18,  11,   5,   1;
  1,   6,  16,  30,  46,  58,  63,  58,  46,  30,  16,   6,   1;
  1,   7,  22,  47,  81, 116, 143, 158, 143, 116,  81,  47,  22,   7,   1;
...
Below are the 21 (3,3) Fibonaccian strings (organized by rank level) that comprise the (3,3)rd symmetric Fibonaccian lattice:
rank=6: (1,4,7)
rank=5: (1,4,8)  (1,5,7)  (2,4,7)
rank=4: (1,4,9)  (1,5,8)  (2,4,8)  (2,5,7)
rank=3: (1,5,9)  (1,6,8)  (2,4,9)  (2,5,8)  (3,5,7)
rank=2: (1,6,9)  (2,5,9)  (2,6,8)  (3,5,8)
rank=1: (2,6,9)  (3,5,9)  (3,6,8)
rank=0: (3,6,9)
The triples (3,4,7), (3,4,8), (3,4,9), (1,6,7), (2,6,7), and (3,6,7) are disallowed as (3,3) Fibonaccian strings since each contains consecutive integers.
In the (3,5)th symmetric Fibonaccian lattice, rank level 8 consists of exactly the (3,5) Fibonaccian strings whose coordinate sum is 3*(5*(5+1)/2)-8=37: (1,4,7,10,15), (1,4,7,11,14), (1,4,8,10,14), (1,4,8,11,13), (1,5,7,10,14), (1,5,7,11,13), (1,5,8,10,13), (2,4,7,10,14), (2,4,7,11,13), (2,4,8,10,13), and (2,5,7,10,13), confirming that T(5,8)=11.

R. G. Donnelly, M. W. Dunkum, M. L. Huber, and L. Knupp, "Sign-alternating Gibonacci polynomials", arXiv:2012.14993 [math.CO], 2020.
R. G. Donnelly, M. W. Dunkum, M. L. Huber, and L. Knupp, "Sign-alternating Gibonacci polynomials", Enumer. Comb. Appl. 1:2 (2021), art. id. S2R15.
R. G. Donnelly and M. W. Dunkum, "Gelfand--Tsetlin-type weight bases for all special linear Lie algebra representations corresponding to skew Schur functions", arXiv:2012.14986 [math.CO], 2020-2022.
R. G. Donnelly and M. W. Dunkum, "Gelfand--Tsetlin-type weight bases for all special linear Lie algebra representations corresponding to skew Schur functions", Adv. Appl. Math. 139 (2022), art. id. 102356.
R. G. Donnelly, M. W. Dunkum, S. V. Lišková, and A. Nance, "Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras", arXiv:2012.14991 [math.CO], 2020-2022.
R. G. Donnelly, M. W. Dunkum, S. V. Lišková, and A. Nance, "Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras", Involve 16:2 (2023), 201-226.

Sum of row n (n >= 0) is A001906(n+1), cf. row n=3 of the array A316269.

With T(0,0)=1, then T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) - T(n-2,k-2) for n >= 1 and 0 <= k <= 2*n, understanding T(i,j) to be zero when j < 0 or j > 2*i. That the preceding recurrence holds is equivalent to the identity p(n,x) = (1+x+x^2)*p(n-1,x) - x^2*p(n-2,x) for n >= 1, where p(0,x)=1 and p(-1,x) is taken to be 0.

A364367 An irregular triangle read by rows, the 4th row-symmetric Fibonaccian triangle: T(n,k) is the Whitney number of level k of the (4,n)-th symmetric Fibonaccian lattice (0 <= n, 0 <= k <= 3*n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 1, 1, 3, 6, 8, 10, 10, 8, 6, 3, 1, 1, 4, 10, 17, 25, 31, 33, 31, 25, 17, 10, 4, 1, 1, 5, 15, 31, 53, 77, 98, 110, 110, 98, 77, 53, 31, 15, 5, 1, 1, 6, 21, 51, 100, 166, 242, 313, 364, 383, 364, 313, 242, 166, 100, 51, 21, 6, 1

Offset: 0

Robert G. Donnelly and Molly W. Dunkum, Jul 20 2023

For integers m and n (m >= 2, n > 0), let L be the set of n-tuples S=(S(1),...,S(n)) with each S(j) in {(j-1)*m+1,(j-1)*m+2,...,j*m} and such that S has no consecutive integers. Partially order these '(m,n) Fibonaccian strings' comprising L by the rule R <= S iff R(j) >= S(j) for 1 <= j <= n (so, 'lightest' n-tuples are at the top of the Hasse diagram for L). Then L is a self-dual distributive lattice, the '(m,n)-th symmetric Fibonaccian lattice'. When n=1, L is a chain with m elements. Now allow n=0; in this case, regard L to be a singleton set. Let p(n,x) be the rank generating function of L, so p(n,1)=|L|, p(0,x)=1, and p(1,x)=1+x+...+x^(m-1). For n >= 2, the fact that p(n,x) = p(1,x)*p(n-1,x) - x^(m-1)*p(n-2,x) can be deduced from a recurrence of Whitney numbers of symmetric Fibonaccian lattices proved in Proposition 2.1 of [Donnelly, Dunkum, Lišková, and Nance, 2023].
The (m,n)-th symmetric Fibonaccian lattice realizes a p(n,1)-dimensional representation of the special linear Lie algebra sl(m,C). The representation is reducible exactly when m >= 3 and n >= 3. The polynomial p(n,x) is a natural specialization of the character of this representation, where the latter can be identified as a certain skew Schur function. In [Donnelly, Dunkum, Lišková, and Nance, 2023], these representations are uniformly constructed (as an application of [Donnelly and Dunkum, 2022]) and explicit formulas for p(n,x) are given.
In [Donnelly, Dunkum, Lišková, and Nance, 2023], the (m,n)-th symmetric Fibonaccian lattice L is also described using semistandard tableaux of a specific ribbon shape; the irreducible components of the associated sl(m,C)-representation are in one-to-one correspondence with what are called the 'ballot-admissible' (aka Littlewood-Richardson) tableaux. In terms of Fibonaccian strings, an element S = (S(1),...,S(n)) in L is ballot-admissible iff for any integer q between 1 and n (inclusive) and any integer r between 1 and m-1 (inclusive), the following integer quantity is nonnegative: Sum_{k=n+1-q..n}([n+1-k is odd]*([r+(k-1)*m = S(k)] - [r+(k-1)*m+1 = S(k)]) + [n+1-k is even]*([k*m-r = S(k)]-[k*m+1-r = S(k)])), where '[]' denotes the Iverson bracket. Enumerating the ballot-admissible tableaux or Fibonaccian strings in L seems to be an interesting problem when m >= 3; when m=3, the sizes of the sets of ballot-admissible tableaux conjecturally agree with A004148.
In this OEIS entry, we have m=4. Let L be the (4,n)-th symmetric Fibonaccian lattice. When n=0, we have T(0,0) = |L| = 1. When n=1, we have T(1,0) = T(1,1) = T(1,2) = T(1,3) = 1 and p(1,x) = 1+x+x^2+x^3, since L is a chain with 4 elements. For n >= 2, we have, by definition, p(n,x) = Sum_{k=0..3*n} T(n,k)*x^k. The Whitney number T(n,k) is the number of (4,n) Fibonaccian strings S=(S(1),...,S(n)) whose coordinate sum S(1)+...+S(n) is equal to 4*(n*(n+1)/2)-k.
For m=3, see A364366. For m=5, see A364368. When m=2, the (2,n)-th symmetric Fibonaccian lattice is a chain with n+1 elements and rank generating function 1+x+...+x^(n-1)+x^n. Therefore, the 2nd row-symmetric Fibonaccian triangle is a regular triangle of 1's. The 1st row-symmetric Fibonaccian 'triangle' is regarded to be the signed sequence 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, ... (A010892). 'Gibonaccian' versions of such triangles are considered in [Donnelly, Dunkum, Huber, and Knupp, 2021].

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) starts as follows:
  1;
  1,   1,   1,   1;
  1,   2,   3,   3,   3,   2,   1;
  1,   3,   6,   8,  10,  10,   8,   6,   3,   1;
  1,   4,  10,  17,  25,  31,  33,  31,  25,  17,  10,   4,   1;
  1,   5,  15,  31,  53,  77,  98, 110, 110,  98,  77,  53,  31,  15,   5,   1;
...
Below are the 15 (4,2) Fibonaccian strings (organized by rank level) that comprise the (4,2)nd symmetric Fibonaccian lattice:
rank=6: (1,5)
rank=5: (1,6)  (2,5)
rank=4: (1,7)  (2,6)  (3,5)
rank=3: (1,8)  (2,7)  (3,6)
rank=2: (2,8)  (3,7)  (4,6)
rank=1: (3,8)  (4,7)
rank=0: (4,8)
The pair (4,5) is disallowed as a (4,2) Fibonaccian string since it contains consecutive integers.
In the (4,3)rd symmetric Fibonaccian lattice, rank level 5 consists of exactly the (4,3) Fibonaccian strings whose coordinate sum is 4*(3*(3+1)/2)-5=19: (1,6,12), (1,7,11), (1,8,10), (2,5,12), (2,6,11), (2,7,10), (3,5,11), (3,6,10), (3,7,9), and (4,6,9), confirming that T(3,5)=10.

R. G. Donnelly, M. W. Dunkum, M. L. Huber, and L. Knupp, "Sign-alternating Gibonacci polynomials", arXiv:2012.14993 [math.CO], 2020.
R. G. Donnelly, M. W. Dunkum, M. L. Huber, and L. Knupp, "Sign-alternating Gibonacci polynomials", Enumer. Comb. Appl. 1:2 (2021), art. id. S2R15.
R. G. Donnelly and M. W. Dunkum, "Gelfand--Tsetlin-type weight bases for all special linear Lie algebra representations corresponding to skew Schur functions", arXiv:2012.14986 [math.CO], 2020-2022.
R. G. Donnelly and M. W. Dunkum, "Gelfand--Tsetlin-type weight bases for all special linear Lie algebra representations corresponding to skew Schur functions", Adv. Appl. Math. 139 (2022), art. id. 102356.
R. G. Donnelly, M. W. Dunkum, S. V. Lišková, and A. Nance, "Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras", arXiv:2012.14991 [math.CO], 2020-2022.
R. G. Donnelly, M. W. Dunkum, S. V. Lišková, and A. Nance, "Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras", Involve 16:2 (2023), 201-226.

Sum of row n (n >= 0) is A001353(n+1), cf. row n=4 of the array A316269.

With T(0,0)=1, then T(n,k) = T(n-1,k-3) + T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) - T(n-2,k-3) for n >= 1 and 0 <= k <= 3*n, understanding T(i,j) to be zero when j < 0 or j > 3*i. That the preceding recurrence holds is equivalent to the identity p(n,x) = (1+x+x^2+x^3)*p(n-1,x) - x^3*p(n-2,x) for n >= 1, where p(0,x)=1 and p(-1,x) is taken to be 0.

A364368 An irregular triangle read by rows, the 5th row-symmetric Fibonaccian triangle: T(n,k) is the Whitney number of level k of the (5,n)-th symmetric Fibonaccian lattice (0 <= n, 0 <= k <= 4*n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 3, 2, 1, 1, 3, 6, 10, 13, 16, 17, 16, 13, 10, 6, 3, 1, 1, 4, 10, 20, 32, 46, 59, 68, 71, 68, 59, 46, 32, 20, 10, 4, 1, 1, 5, 15, 35, 66, 109, 161, 215, 263, 296, 308, 296, 263, 215, 161, 109, 66, 35, 15, 5, 1

Offset: 0

Robert G. Donnelly and Molly W. Dunkum, Jul 20 2023

For integers m and n (m >= 2, n > 0), let L be the set of n-tuples S=(S(1),...,S(n)) with each S(j) in {(j-1)*m+1,(j-1)*m+2,...,j*m} and such that S has no consecutive integers. Partially order these '(m,n) Fibonaccian strings' comprising L by the rule R <= S iff R(j) >= S(j) for 1 <= j <= n (so, 'lightest' n-tuples are at the top of the Hasse diagram for L). Then L is a self-dual distributive lattice, the '(m,n)-th symmetric Fibonaccian lattice'. When n=1, L is a chain with m elements. Now allow n=0; in this case, regard L to be a singleton set. Let p(n,x) be the rank generating function of L, so p(n,1)=|L|, p(0,x)=1, and p(1,x)=1+x+...+x^(m-1). For n >= 2, the fact that p(n,x) = p(1,x)*p(n-1,x) - x^(m-1)*p(n-2,x) can be deduced from a recurrence of Whitney numbers of symmetric Fibonaccian lattices proved in Proposition 2.1 of [Donnelly, Dunkum, Lišková, and Nance, 2023].
The (m,n)-th symmetric Fibonaccian lattice realizes a p(n,1)-dimensional representation of the special linear Lie algebra sl(m,C). The representation is reducible exactly when m >= 3 and n >= 3. The polynomial p(n,x) is a natural specialization of the character of this representation, where the latter can be identified as a certain skew Schur function. In [Donnelly, Dunkum, Lišková, and Nance, 2023], these representations are uniformly constructed (as an application of [Donnelly and Dunkum, 2022]) and explicit formulas for p(n,x) are given.
In [Donnelly, Dunkum, Lišková, and Nance, 2023], the (m,n)-th symmetric Fibonaccian lattice L is also described using semistandard tableaux of a specific ribbon shape; the irreducible components of the associated sl(m,C)-representation are in one-to-one correspondence with what are called the 'ballot-admissible' (aka Littlewood-Richardson) tableaux. In terms of Fibonaccian strings, an element S = (S(1),...,S(n)) in L is ballot-admissible iff for any integer q between 1 and n (inclusive) and any integer r between 1 and m-1 (inclusive), the following integer quantity is nonnegative: Sum_{k=n+1-q..n}([n+1-k is odd]*([r+(k-1)*m = S(k)] - [r+(k-1)*m+1 = S(k)]) + [n+1-k is even]*([k*m-r = S(k)]-[k*m+1-r = S(k)])), where '[]' denotes the Iverson bracket. Enumerating the ballot-admissible tableaux or Fibonaccian strings in L seems to be an interesting problem when m >= 3; when m=3, the sizes of the sets of ballot-admissible tableaux conjecturally agree with A004148.
In this OEIS entry, we have m=5. Let L be the (5,n)-th symmetric Fibonaccian lattice. When n=0, we have T(0,0) = |L| = 1. When n=1, we have T(1,0) = T(1,1) = T(1,2) = T(1,3) = T(1,4) = 1 and p(1,x) = 1+x+x^2+x^3+x^4, since L is a chain with 5 elements. For n >= 2, we have, by definition, p(n,x) = Sum_{k=0..4*n} T(n,k)*x^k. The Whitney number T(n,k) is the number of (5,n) Fibonaccian strings S=(S(1),...,S(n)) whose coordinate sum S(1)+...+S(n) is equal to 5*(n*(n+1)/2)-k.
For m=3, see A364366. For m=4, see A364367. When m=2, the (2,n)-th symmetric Fibonaccian lattice is a chain with n+1 elements and rank generating function 1+x+...+x^(n-1)+x^n. Therefore, the 2nd row-symmetric Fibonaccian triangle is a regular triangle of 1's. The 1st row-symmetric Fibonaccian 'triangle' is regarded to be the signed sequence 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, ... (A010892). 'Gibonaccian' versions of such triangles are considered in [Donnelly, Dunkum, Huber, and Knupp, 2021].

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) starts as follows:
  1;
  1,  1,  1,  1,  1;
  1,  2,  3,  4,  4,  4,  3,  2,  1;
  1,  3,  6, 10, 13, 16, 17, 16, 13, 10,  6,  3,  1;
  1,  4, 10, 20, 32, 46, 59, 68, 71, 68, 59, 46, 32, 20, 10,  4,  1;
...
Below are the 24 (5,2) Fibonaccian strings (organized by rank level) that comprise the (5,2)nd symmetric Fibonaccian lattice:
rank=8: (1,6)
rank=7: (1,7)   (2,6)
rank=6: (1,8)   (2,7)   (3,6)
rank=5: (1,9)   (2,8)   (3,7)   (4,6)
rank=4: (1,10)  (2,9)   (3,8)   (4,7)
rank=3: (2,10)  (3,9)   (4,8)   (5,7)
rank=2: (3,10)  (4,9)   (5,8)
rank=1: (4,10)  (5,9)
rank=0: (5,10)
The pair (5,6) is disallowed as a (5,2) Fibonaccian string since it contains consecutive integers.
In the (5,3)rd symmetric Fibonaccian lattice, rank level 9 consists of exactly the (5,3) Fibonaccian strings whose coordinate sum is 5*(3*(3+1)/2)-9=21: (1,6,14), (1,7,13), (1,8,12), (1,9,11), (2,6,13), (2,7,12), (2,8,11), (3,6,12), (3,7,11), and (4,6,11), confirming that T(3,9)=10.

R. G. Donnelly, M. W. Dunkum, M. L. Huber, and L. Knupp, Sign-alternating Gibonacci polynomials, arXiv:2012.14993 [math.CO], 2020.
R. G. Donnelly, M. W. Dunkum, M. L. Huber, and L. Knupp, "Sign-alternating Gibonacci polynomials", Enumer. Comb. Appl. 1:2 (2021), art. id. S2R15.
R. G. Donnelly and M. W. Dunkum, Gelfand--Tsetlin-type weight bases for all special linear Lie algebra representations corresponding to skew Schur functions, arXiv:2012.14986 [math.CO], 2020-2022.
R. G. Donnelly and M. W. Dunkum, "Gelfand--Tsetlin-type weight bases for all special linear Lie algebra representations corresponding to skew Schur functions", Adv. Appl. Math. 139 (2022), art. id. 102356.
R. G. Donnelly, M. W. Dunkum, S. V. Lišková, and A. Nance, Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras, arXiv:2012.14991 [math.CO], 2020-2022.
R. G. Donnelly, M. W. Dunkum, S. V. Lišková, and A. Nance, "Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras", Involve 16:2 (2023), 201-226.

Sum of row n (n >= 0) is A004254(n+1), cf. row n=5 of the array A316269.

With T(0,0)=1, then T(n,k) = T(n-1,k-4) + T(n-1,k-3) + T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) - T(n-2,k-4) for n >= 1 and 0 <= k <= 4*n, understanding T(i,j) to be zero when j < 0 or j > 4*i. That the preceding recurrence holds is equivalent to the identity p(n,x) = (1+x+x^2+x^3+x^4)*p(n-1,x) - x^4*p(n-2,x) for n >= 1, where p(0,x)=1 and p(-1,x) is taken to be 0.

A365691 G.f. satisfies A(x) = 1 + x^2A(x)^5 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 1, 1, 6, 12, 54, 147, 593, 1886, 7292, 25204, 96153, 348304, 1327716, 4946471, 18936366, 71827598, 276612103, 1062220253, 4115807184, 15947902376, 62148513732, 242485933208, 949828266722, 3726623622402, 14663689944397, 57798199213989

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A004148, A005043, A025246, A046736, A365690.

Cf. A365693.

a(n) = sum(k=0, n\2, binomial(n-k-1, n-2*k)*binomial(n+3*k+1, k)/(n+3*k+1));

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

A023428 Generalized Catalan Numbers x^4A(x)^2 -(1-x+x^4+x^5)A(x) +1 =0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 17, 27, 45, 77, 132, 224, 378, 640, 1093, 1881, 3250, 5622, 9732, 16874, 29332, 51126, 89313, 156283, 273842, 480474, 844220, 1485472, 2617335, 4617243, 8154289, 14415869, 25511256, 45190366, 80124434, 142189496

Offset: 0

Olivier Gérard

Cf. A000108, A001006, A004148, A006318.

A023428 := proc(n)
    option remember;
    if n = 0 then
        1 ;
    else
        procname(n-1)+add(procname(k)*procname(n-4-k),k=2..n-4) ;
    end if;
end proc:
seq(A023428(n),n=0..80) ; # R. J. Mathar, Oct 31 2014

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 2, n-4} ];

a(0)=1; a(n) = a(n-1) + Sum_{k=2..n-4} a(k)*a(n-4-k).

G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x + x^4 + x^5). - Ilya Gutkovskiy, Jul 20 2021

More terms from Sean A. Irvine, Jun 04 2019

A023429 Generalized Catalan Numbers x^4A(x)^2 -(1-x+x^4+x^5+x^6)A(x) + 1 =0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 17, 27, 44, 73, 122, 204, 340, 566, 945, 1586, 2674, 4521, 7656, 12982, 22047, 37509, 63934, 109166, 186685, 319679, 548091, 940819, 1616830, 2781706, 4790949, 8259748, 14253480, 24618561, 42557378

Offset: 0

Olivier Gérard

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000108, A001006, A004148, A006318.

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-4-k), k=3..n-4))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 07 2011

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 3, n-4} ];

G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x + x^4 + x^5 + x^6). - Ilya Gutkovskiy, Jul 20 2021

A023430 Generalized Catalan Numbers x^4A(x)^2 -(1-x+x^4+x^5+x^6+x^7)A(x) + 1 =0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 17, 27, 44, 72, 118, 194, 320, 528, 871, 1439, 2385, 3965, 6605, 11017, 18399, 30771, 51538, 86440, 145165, 244085, 410890, 692442, 1168114, 1972470, 3333834, 5639888, 9549311, 16181931, 27442827, 46575013

Offset: 0

Olivier Gérard

Cf. A000108, A001006, A004148, A006318.

A023430 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        procname(n-1)+add(procname(k)*procname(n-4-k),k=4..n-4) ;
    end if;
end proc: # R. J. Mathar, May 01 2015

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 4, n-4} ];

G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x + x^4 + x^5 + x^6 + x^7). - Ilya Gutkovskiy, Jul 20 2021

More terms from Sean A. Irvine, Jun 04 2019

cp's OEIS Frontend

A307733 a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-1} a(k) * a(n-k-1).

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A329672 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps UU.

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A329674 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps DD.

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A364366 An irregular triangle read by rows, the 3rd row-symmetric Fibonaccian triangle: T(n,k) is the Whitney number of level k of the (3,n)-th symmetric Fibonaccian lattice (0 <= n, 0 <= k <= 2*n).

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A364367 An irregular triangle read by rows, the 4th row-symmetric Fibonaccian triangle: T(n,k) is the Whitney number of level k of the (4,n)-th symmetric Fibonaccian lattice (0 <= n, 0 <= k <= 3*n).

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A364368 An irregular triangle read by rows, the 5th row-symmetric Fibonaccian triangle: T(n,k) is the Whitney number of level k of the (5,n)-th symmetric Fibonaccian lattice (0 <= n, 0 <= k <= 4*n).

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A365691 G.f. satisfies A(x) = 1 + x^2*A(x)^5 / (1 - x*A(x)).

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A023428 Generalized Catalan Numbers x^4*A(x)^2 -(1-x+x^4+x^5)*A(x) +1 =0.

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A023429 Generalized Catalan Numbers x^4*A(x)^2 -(1-x+x^4+x^5+x^6)*A(x) + 1 =0.

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Maple

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A365691 G.f. satisfies A(x) = 1 + x^2A(x)^5 / (1 - xA(x)).

A023428 Generalized Catalan Numbers x^4A(x)^2 -(1-x+x^4+x^5)A(x) +1 =0.

A023429 Generalized Catalan Numbers x^4A(x)^2 -(1-x+x^4+x^5+x^6)A(x) + 1 =0.

A023430 Generalized Catalan Numbers x^4A(x)^2 -(1-x+x^4+x^5+x^6+x^7)A(x) + 1 =0.