A105633 -id:A105633 - cp's OEIS frontend

A346077 a(n) = 1 + Sum_{k=1..n-5} a(k) * a(n-k-5).

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 12, 18, 26, 36, 49, 69, 101, 150, 221, 320, 460, 667, 981, 1456, 2161, 3191, 4698, 6932, 10283, 15324, 22870, 34103, 50813, 75770, 113229, 169590, 254340, 381579, 572537, 859511, 1291681, 1943489, 2926980, 4410709, 6649220, 10028570

Offset: 0

Ilya Gutkovskiy, Jul 04 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A105633, A217282, A307972, A346049, A346074, A346075, A346076.

a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 5], {k, 1, n - 5}]; Table[a[n], {n, 0, 47}]
nmax = 47; A[] = 0; Do[A[x] = 1/(1 - x) + x^5 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

@CachedFunction
def a(n): # a = A346077
    if (n<6): return 1
    else: return 1 + sum(a(k)*a(n-k-5) for k in range(1,n-4))
[a(n) for n in range(51)] # G. C. Greubel, Nov 27 2022

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

A377441 Square array T(n, k) read by rising antidiagonals. Row n has the ordinary generating function (-(nx^3-(n+1)x^2+x) + sqrt((nx^3-(n+1)x^2+x)^2 - 4(x^3-x^2)((n+1)x^2-x)))/(2(x^3-x^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 6, 14, 1, 1, 2, 7, 21, 42, 1, 1, 2, 8, 30, 78, 132, 1, 1, 2, 9, 41, 136, 299, 429, 1, 1, 2, 10, 54, 222, 630, 1172, 1430, 1, 1, 2, 11, 69, 342, 1221, 2959, 4677, 4862, 1, 1, 2, 12, 86, 502, 2192, 6774, 14058, 18947, 16796, 1, 1, 2, 13, 105, 708, 3687, 14129, 37853, 67472, 77746, 58786, 1, 1, 2, 14, 126, 966, 5874, 27184

Offset: 0

Thomas Scheuerle, Oct 28 2024

The Hankel sequence transform of row n satisfies the Somos-4 recurrence c(k) = (c(k-1) * c(k-3) + n*c(k-2)^2) / c(k-4). All Somos-4 sequences which are beginning with 1, 1, 1, 1, n, ... will be covered, but the Hankel transform will start with the terms 1, n, ... in each case.

			The array begins:
  [0] 1, 1, 2,  5, 14,  42,  132,   429,   1430, ... = A000108
  [1] 1, 1, 2,  6, 21,  78,  299,  1172,   4677, ... = A254316
  [2] 1, 1, 2,  7, 30, 136,  630,  2959,  14058, ...
  [3] 1, 1, 2,  8, 41, 222, 1221,  6774,  37853, ...
  [4] 1, 1, 2,  9, 54, 342, 2192, 14129,  91494, ...
  [5] 1, 1, 2, 10, 69, 502, 3687, 27184, 201045, ...

Cf. A377442 (extension for -n), A105633 (row -1), A152172 (row -2).
Cf. A000108 (row 0), A254316 (row 1).
Cf. A000012 (Hankel transform of row 0), A006720 (Hankel transform of row 1).
Cf. A330025 (Hankel transform of row -1), A328380 (Hankel transform of row -2).
Cf. A371965, A377443.

T(n, max_k) = Vec(-2*((n+1)*x-1)/((x-1)*(n*x-1)+((n*x^2-(n+1)*x+1)^2-4*x*(x-1)*((n+1)*x-1)+O(x^max_k))^(1/2)))

The generating function A(x) of row n satisfies: 0 = (x^3 - x^2)*A(x)^2 + (n*x^3 - (n+1)*x^2 - x)*A(x) + ((n+1)*x^2 - x).
Let d(m, n) = ( d(m-3, n)*d(m-2, n) + n)/( d(m-5, n)*d(m-4, n)*d(m-3, n)^2*d(m-2, n)^2*d(m-1, n) ) for m = even and d(m, n) = 1/( d(m-1, n)*d(m-2, n) ) for m = odd with d( < 1 , n) = 1, then the generating function of row n can be expanded as continued fractions: 1/(1 - x/(1 - d(0, n)*x/(1 - d(1, n)*x/(1 - d(2, n)*x/(...))))).
d(m, n)*d(m+1, n) is a rational solution in x of the elliptic equation y^2 = -4*x^3 + ((n+1)^2 + 8)*x^2 - 2*(n+3)*x + 1. The division polynomials for multiples of the point with x = 1, correspondent to the Hankel transform of row n in the array T(n, k).
T(n, k + 2) = Sum_{j >= 0} A377443(k, j)*n^j. This polynomial starts with A000108(k+2) + A371965(k+2)*n + ..., where A371965 is known to count peaks in the set of Catalan words of length k.

A057580 Maximal numbers of codewords in mixed code with 2 binary coordinates and n ternary coordinates with Hamming distance 3.

Original entry on oeis.org

1, 2, 4, 9, 22

Offset: 0

N. J. A. Sloane, Oct 04 2000

Is this sequence equal to A105633? - Paul D. Hanna, Apr 17 2005

Patric R. J. Östergård, Classification of small binary/ternary one-error-correcting codes (Appendix), Discrete Mathematics 223 (2000), 253-262.

Cf. A057574-A057584, A050142.

A105849 Row sums of number triangle A105848.

Original entry on oeis.org

1, 3, 9, 28, 91, 308, 1079, 3888, 14332, 53810, 205075, 791250, 3084504, 12129506, 48056095, 191633546, 768535768, 3097705378, 12541851048, 50983349848, 208003171266, 851412895348, 3495527318559, 14390543072502, 59393240482618

Offset: 0

Paul Barry, Apr 22 2005

Binomial transform of A105633.

G.f.: (1-2x-sqrt((1-6x+8x^2-4x^3)/(1-2x)))/(2x^2).

Conjecture: (n+2)*a(n) +2*(-4*n-3)*a(n-1) +4*(5*n-2)*a(n-2) +2*(-10*n+17)*a(n-3) +8*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 16 2012

A272794 The numbers of closed simply typable lambda terms of natural size n.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 13, 27, 74, 198, 508, 1371, 3809, 10477, 29116, 82419, 233748, 666201, 1914668, 5528622, 16019330, 46642245, 136326126, 399652720, 1175422931, 3467251920, 10258152021

Offset: 0

Pierre Lescanne, Jul 13 2016

Natural size measure lambda terms as follows: all symbols are assigned size 1, namely applications, abstractions, successor symbols in de Bruijn indices and 0 symbol in de Bruijn indices (i.e., a de Bruijn index n is assigned size n+1).
Here we count the closed simply typable terms of natural size n. "Closed" means that there is no free index (no free bound variable). "Simply typable" means that lambda terms have a simple type.
The numbers are computed as follows: all the closed terms are generated and then filtered using a type reconstruction algorithm. The values given above are the only known values of the sequence.

Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, A natural counting of Lambda terms, arXiv:1506.02367 [cs.LO], 2015.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, A Natural Counting of Lambda Terms, SOFSEM 2016: 183-194
Maciej Bendkowski, K Grygiel, P Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682, 2016

Cf. A105633, A220471, A236393, A236405.

A275057 Numbers of closed lambda terms of natural size n.

Original entry on oeis.org

0, 0, 1, 1, 3, 6, 17, 41, 116, 313, 895, 2550, 7450, 21881, 65168, 195370, 591007, 1798718, 5510023, 16966529, 52506837, 163200904, 509323732, 1595311747, 5013746254, 15805787496, 49969942138, 158396065350, 503317495573, 1602973785463, 5116010587910, 16360492172347

Offset: 0

Pierre Lescanne, Jul 14 2016

nonn

Natural size measure lambda terms as follows: all symbols are assigned size 1, namely applications, abstractions, successor symbols in de Bruijn indices and 0 symbol in de Bruijn indices (i.e., a de Bruijn index n is assigned size n+1).

Here we count the closed terms of natural size n, where "closed" means that there is no free index (no free bound variable).

Pierre Lescanne, Table of n, a(n) for n = 0..299
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, A Natural Counting of Lambda Terms, SOFSEM 2016: 183-194.
Maciej Bendkowski, K Grygiel, P Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682 [cs.LO], 2016-2017.

Cf. A105633, A272794.

L[0, ] = 0; L[n, m_] := L[n, m] = Sum[L[k, m]*L[n-k-1, m], {k, 0, n-1}] + L[n-1, m+1] + Boole[m >= n];
a[n_] := L[n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, May 23 2017 *)

L(0,m) = 0.

L(n+1,m) = (Sum_{k=0..n} L(k,m)*L(n-k,m)) + L(n,m+1) + [m >= n+1], where [p(n,m)] = 1 if p(n,m) is true and [p(n,m)] = 0 if p(n,m) is false then one considers the sequence (L(n,0)).

A377442 Square array read by rising antidiagonals: T(n, k) = A377441(-n, k), an extension of A377441 into the domain of negative n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 4, 14, 1, 1, 2, 3, 9, 42, 1, 1, 2, 2, 6, 22, 132, 1, 1, 2, 1, 5, 12, 57, 429, 1, 1, 2, 0, 6, 6, 26, 154, 1430, 1, 1, 2, -1, 9, -2, 15, 59, 429, 4862, 1, 1, 2, -2, 14, -18, 24, 24, 138, 1223, 16796, 1, 1, 2, -3, 21, -48, 77, -23, 53, 332, 3550, 58786, 1, 1, 2, -4, 30, -98, 222, -226, 102, 107, 814, 10455, 208012, 1, 1, 2

Offset: 0

Thomas Scheuerle, Nov 04 2024

The main entry for this array is A377441.

			The array begins:
  [ 0] 1, 1, 2,  5, 14,  42,  132,   429,   1430, ... = A000108
  [-1] 1, 1, 2,  4,  9,  22,   57,   154,    429, ... = A105633
  [-2] 1, 1, 2,  3,  6,  12,   26,    59,    138, ... = A152172
  [-3] 1, 1, 2,  2,  5,   6,   15,    24,     53, ...
  [-4] 1, 1, 2,  1,  6,  -2,   24,   -23,    102, ...
  [-5] 1, 1, 2,  0,  9, -18,   77,  -226,    765, ...
  [-6] 1, 1, 2, -1, 14, -48,  222,  -921,   3914, ...
  [-7] 1, 1, 2, -2, 21, -98,  531, -2756,  14373, ...
Row index written as [m] is corresponding to A377441(m, k).

Cf. A377441 (The main entry for this sequence).
Cf. A105633 (row -1), A152172 (row -2).
Cf. A000108 (row 0), A254316 (row 1).
Cf. A000012 (Hankel transform of row 0), A006720 (Hankel transform of row 1).
Cf. A330025 (Hankel transform of row -1), A328380 (Hankel transform of row -2).
Cf. A371965, A377443.

A274490 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n starting with k columns of length 1 (n>=2, k>=0).

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 3, 1, 0, 1, 22, 8, 3, 1, 0, 1, 62, 22, 8, 3, 1, 0, 1, 178, 62, 22, 8, 3, 1, 0, 1, 519, 178, 62, 22, 8, 3, 1, 0, 1, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1, 13800, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1

Offset: 2

Emeric Deutsch, Jun 25 2016

Number of entries in row n is n.
Sum of entries in row n = A082582(n).
T(n,0) = A188464(n-3) (n>=3).
Sum_{k>=0} k*T(n,k) = A105633(n-2).

			Row 4 is 3,1,0,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, they start with 3, 1, 0, 0, 0 columns of length 1.
Triangle starts
0,1;
1,0,1;
3,1,0,1;
8,3,1,0,1;
22,8,3,1,0,1

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Cf. A082582, A105633, A188464.

G := (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form

nmax = 12;
g = (1 - 3z + z^2 + 2t z^3 - z^3 - (1-z) Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/ (2z (1 - t z));
cc = CoefficientList[g + O[z]^(nmax+1), z];
T[n_, k_] := Coefficient[cc[[n+1]], t, k];
Table[T[n, k], {n, 2, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)

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

A294450 The numbers of plain simply typable lambda terms of natural size n.

Original entry on oeis.org

0, 1, 2, 3, 8, 17, 42, 106, 287, 747, 2069, 5732, 16012, 45283, 129232, 370761, 1069972

Offset: 0

N. J. A. Sloane, Nov 22 2017

Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, A natural counting of Lambda terms, arXiv:1506.02367 [cs.LO], 2015.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, A Natural Counting of Lambda Terms, SOFSEM 2016: 183-194.
Maciej Bendkowski, K. Grygiel, P. Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682 [cs.LO], 2016-2017.

Cf. A105633, A220471, A236393, A236405, A272794.

cp's OEIS Frontend

A346077 a(n) = 1 + Sum_{k=1..n-5} a(k) * a(n-k-5).

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A377441 Square array T(n, k) read by rising antidiagonals. Row n has the ordinary generating function (-(n*x^3-(n+1)*x^2+x) + sqrt((n*x^3-(n+1)*x^2+x)^2 - 4*(x^3-x^2)*((n+1)*x^2-x)))/(2*(x^3-x^2)).

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A057580 Maximal numbers of codewords in mixed code with 2 binary coordinates and n ternary coordinates with Hamming distance 3.

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A105849 Row sums of number triangle A105848.

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A272794 The numbers of closed simply typable lambda terms of natural size n.

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A275057 Numbers of closed lambda terms of natural size n.

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A377442 Square array read by rising antidiagonals: T(n, k) = A377441(-n, k), an extension of A377441 into the domain of negative n.

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A274490 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n starting with k columns of length 1 (n>=2, k>=0).

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A294450 The numbers of plain simply typable lambda terms of natural size n.

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A377441 Square array T(n, k) read by rising antidiagonals. Row n has the ordinary generating function (-(nx^3-(n+1)x^2+x) + sqrt((nx^3-(n+1)x^2+x)^2 - 4(x^3-x^2)((n+1)x^2-x)))/(2(x^3-x^2)).