A040000 -id:A040000 - cp's OEIS frontend

A225094 Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 6, 2, 0, 1, 1, 24, 54, 2, 0, 1, 1, 120, 1944, 384, 2, 0, 1, 1, 720, 99000, 132000, 2550, 2, 0, 1, 1, 5040, 6966000, 79716000, 8059800, 16506, 2, 0, 1, 1, 40320, 655678800, 78928416000, 57010275000, 471369024, 105840, 2, 0, 1

Offset: 0

Alois P. Heinz, Apr 27 2013

An interior point p = (p_1, ..., p_k) has k>0 components with 0

			A(n,0) = 1: [()].
A(0,k) = 1: [{0}^k].
A(1,1) = 1: [(1), (0)].
A(2,1) = 0, there is no path from (2) to (0) without interior points.
A(1,2) = 2: [(1,1), (0,1), (0,0)], [(1,1), (1,0), (0,0)].
A(1,3) = 6: [(1,1,1), (0,1,1), (0,0,1), (0,0,0)], [(1,1,1), (0,1,1), (0,1,0), (0,0,0)], [(1,1,1), (1,0,1), (0,0,1), (0,0,0)], [(1,1,1), (1,0,1), (1,0,0), (0,0,0)], [(1,1,1), (1,1,0), (0,1,0), (0,0,0)], [(1,1,1), (1,1,0), (1,0,0), (0,0,0)].
Square array A(n,k) begins:
  1, 1, 1,     1,         1,              1, ...
  1, 1, 2,     6,        24,            120, ...
  1, 0, 2,    54,      1944,          99000, ...
  1, 0, 2,   384,    132000,       79716000, ...
  1, 0, 2,  2550,   8059800,    57010275000, ...
  1, 0, 2, 16506, 471369024, 38606650125120, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0, 2-4 give: A000012, A040000, A060774, A225220.
Rows n=0-4 give: A000012, A000142, A071798(k) (for k>0), A225096, A225221.
Main diagonal gives: A225111.
Cf. A089759 (unrestricted paths), A210472, A262809, A263159.

b:= proc(n, l) option remember; local m; m:= nops(l);
      `if`(m=0 or l[m]=0, 1, `if`(l[1]>0 and l[m] b(n, [n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, l_] := b[n, l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[l[[1]] > 0 && l[[m]] < n, 0, Sum[If[l[[i]] == 0, 0, b[n, Sort[ReplacePart[l, i -> l[[i]] - 1]]]], {i, 1, m}]]] ]; a[n_, k_] := b[n, Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A080934 Square array read by antidiagonals of number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2n steps with all values less than or equal to k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 8, 1, 0, 1, 1, 2, 5, 13, 16, 1, 0, 1, 1, 2, 5, 14, 34, 32, 1, 0, 1, 1, 2, 5, 14, 41, 89, 64, 1, 0, 1, 1, 2, 5, 14, 42, 122, 233, 128, 1, 0, 1, 1, 2, 5, 14, 42, 131, 365, 610, 256, 1, 0, 1, 1, 2, 5, 14, 42, 132, 417, 1094, 1597, 512, 1, 0

Offset: 0

Henry Bottomley, Feb 25 2003

Number of permutations in S_n avoiding both 132 and 123...k.
T(n,k) = number of rooted ordered trees on n nodes of depth <= k. Also, T(n,k) = number of {1,-1} sequences of length 2n summing to 0 with all partial sums are >=0 and <= k. Also, T(n,k) = number of closed walks of length 2n on a path of k nodes starting from (and ending at) a node of degree 1. - Mitch Harris, Mar 06 2004
Also T(n,k) = k-th coefficient in expansion of the rational function R(n), where R(1) = 1, R(n+1) = 1/(1-x*R(n)), which means also that lim(n->inf,R(n)) = g.f. of Catalan numbers (A000108) wherever it has real value (see Mansour article). - Clark Kimberling and Ralf Stephan, May 26 2004
Row n of the array gives Taylor series expansion of F_n(t)/F_{n+1}(t), where F_n(t) are the Fibonacci polynomials defined in A259475 [Kreweras, 1970]. - N. J. A. Sloane, Jul 03 2015

			T(3,2) = 4 since the paths of length 2*3 (7 points) with all values less than or equal to 2 can take the routes 0101010, 0101210, 0121010 or 0121210, but not 0123210.
From _Peter Luschny_, Aug 27 2014: (Start)
Trees with n nodes and height <= h:
h\n  1  2  3  4   5   6    7    8     9    10     11
---------------------------------------------------------
[ 1] 1, 0, 0, 0,  0,  0,   0,   0,    0,    0,     0, ...  A063524
[ 2] 1, 1, 1, 1,  1,  1,   1,   1,    1,    1,     1, ...  A000012
[ 3] 1, 1, 2, 4,  8, 16,  32,  64,  128,  256,   512, ...  A011782
[ 4] 1, 1, 2, 5, 13, 34,  89, 233,  610, 1597,  4181, ...  A001519
[ 5] 1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281,  9842, ...  A124302
[ 6] 1, 1, 2, 5, 14, 42, 131, 417, 1341, 4334, 14041, ...  A080937
[ 7] 1, 1, 2, 5, 14, 42, 132, 428, 1416, 4744, 16016, ...  A024175
[ 8] 1, 1, 2, 5, 14, 42, 132, 429, 1429, 4846, 16645, ...  A080938
[ 9] 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861, 16778, ...  A033191
[10] 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, ...  A211216
---------------------------------------------------------
The generating functions are listed in A211216. Note that the values up to the main diagonal are the Catalan numbers A000108.
(End)

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Pattern-restricted cyclic permutations with a pattern-restricted cycle form, arXiv:2408.15000 [math.CO], 2024. See also Enum. Comb. Appl. (2025) Vol. 5, No. 1, Art. No. S2R3. See p. 7.
Nicolaas Govert de Bruijn, Donald E. Knuth, and Stephen O. Rice, The Average Height of Planted Plane Trees, Graph Theory and Computing, (1972) 15-22.
Yann Dijoux, Chebyshev polynomials involved in the Householder's method for square roots, arXiv:2501.04703 [math.GM], 2024. Mentions this sequence.
Andrzej Dudek, Jarosław Grytczuk, and Andrzej Ruciński, Largest bipartite sub-matchings of a random ordered matching or a problem with socks, arXiv:2402.02223 [math.CO], 2024.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Aleksandar Ilic and Andreja Ilic, On the number of restricted Dyck paths, Filomat 25:3 (2011), 191-201; DOI: 10.2298/FIL1103191I.
Anthony Joseph and Polyxeni Lamprou, A new interpretation of Catalan numbers, arXiv preprint arXiv:1512.00406 [math.CO], 2015.
Myrto Kallipoliti, Robin Sulzgruber, and Eleni Tzanaki, Patterns in Shi tableaux and Dyck paths, arXiv:2006.06949 [math.CO], 2020.
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (page 10, Corollary 3).
Christian Krattenthaler, Permutations with restricted patterns and Dyck paths, arXiv:math/0002200 [math.CO], 2000.
Germain Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41. See page 38.
Germain Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
Erkko Lehtonen and Tamás Waldhauser, Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs, arXiv:2011.07621 [math.CO], 2020. See also J. of Algebraic Combinatorics (2021) Vol. 53, 613-638.
Toufik Mansour, Restricted even permutations and Chebyshev polynomials, arXiv:math/0302014 [math.CO], 2003.
Toufik Mansour and Alek Vainshtein, Restricted permutations, continued fractions and Chebyshev polynomials, arXiv:math/9912052 [math.CO], 1999-2000.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See pp. 5-6.
Thomas Tunstall, Tim Rogers, and Wolfram Möbius, Assisted percolation of slow-spreading mutants in heterogeneous environments, arXiv:2303.01579 [q-bio.PE], 2023.
Vince White, Enumeration of Lattice Paths with Restrictions, (2024). Electronic Theses and Dissertations. 2799. See pp. 20, 25.

Cf. A000108, A079214, A080935, A080936. Rows include A000012, A057427, A040000 (offset), columns include (essentially) A000007, A000012, A011782, A001519, A007051, A080937, A024175, A080938, A033191, A211216. Main diagonal is A000108.

Cf. A094718 (involutions). Cf. also A259475.

# As a triangular array:
b:= proc(x, y, k) option remember; `if`(y>min(k, x) or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, k)+ b(x-1, y+1, k)))
    end:
A:= (n, k)-> b(2*n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 06 2012
# As a square array:
A := proc(n,k) option remember; local j; if n = 1 then 1 elif k = 1 then 0 else add(A(n-j,k)*A(j,k-1), j=1..n-1) fi end:
linalg[matrix](10, 12, (n,k) -> A(k,n)); # Peter Luschny, Aug 27 2014

A[n_, k_] := A[n, k] = Which[n == 1, 1, k == 1, 0, True, Sum[A[n-j, k]*A[j, k-1], {j, 1, n-1}]]; Table[A[k-n+1, n], {k, 1, 13}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Peter Luschny *)

A(N, K) = {
  my(m = matrix(N, K, n, k, n==1));
  for (n = 2, N,
  for (k = 2, K,
       m[n,k] = sum(i = 1, n-1, m[n-i,k] * m[i,k-1])));
  return(m);
}
A(11,10)~  \\ Gheorghe Coserea, Jan 13 2016

T(n, k) = Sum_{0A080935(n, k) = T(n, k-1)+A080936(n, k); for k>=n T(n, k) = A000108(n).
T(n, k) = 2^(2n+1)/(k+2) * Sum_{i=1..k+1} (sin(Pi*i/(k+2))*cos(Pi*i/(k+2))^n)^2 for n>=1. - Herbert Kociemba, Apr 28 2004
G.f. of n-th row: B(n)/B(n+1) where B(j)=[(1+sqrt(1-4x))/2]^j-[(1-sqrt(1-4x))/2]^j.

A104698 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k, j)*binomial(n-j+1, k+1).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 6, 1, 5, 16, 19, 8, 1, 6, 25, 44, 33, 10, 1, 7, 36, 85, 96, 51, 12, 1, 8, 49, 146, 225, 180, 73, 14, 1, 9, 64, 231, 456, 501, 304, 99, 16, 1, 10, 81, 344, 833, 1182, 985, 476, 129, 18, 1, 11, 100, 489, 1408, 2471, 2668, 1765, 704, 163, 20, 1, 12

Offset: 0

Gary W. Adamson, Mar 19 2005

The n-th column of the triangle is the binomial transform of the n-th row of A081277, followed by zeros. Example: column 3, (1, 6, 19, 44, ...) = binomial transform of row 3 of A081277: (1, 5, 8, 4, 0, 0, 0, ...). A104698 = reversal by rows of A142978. - Gary W. Adamson, Jul 17 2008
This sequence is jointly generated with A210222 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) + v(n-1) + 1 and v(n,x) = 2x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section at A210222. - Clark Kimberling, Mar 19 2012
This Riordan triangle T appears in a formula for A001100(n, 0) = A002464(n), for n >= 1. - Wolfdieter Lang, May 13 2025

			The Riordan triangle T begins:
  n\k  0   1   2    3    4    5    6   7   8  9 10 ...
  ----------------------------------------------------
  0:   1
  1:   2   1
  2:   3   4   1
  3:   4   9   6    1
  4:   5  16  19    8    1
  5:   6  25  44   33   10    1
  6:   7  36  85   96   51   12    1
  7:   8  49 146  225  180   73   14   1
  8:   9  64 231  456  501  304   99  16   1
  9:  10  81 344  833 1182  985  476 129  18  1
  10: 11 100 489 1408 2471 2668 1765 704 163 20  1
  ... reformatted and extended by _Wolfdieter Lang_, May 13 2025
From _Wolfdieter Lang_, May 13 2025: (Start)
Zumkeller recurrence (adapted for offset [0,0]): 19 = T(4, 2) = T(2, 1) + T(3, 1) + T(3,3) = 4 + 9 + 6 = 19.
A-sequence recurrence: 19 = T(4, 2) = 1*T(3. 1) + 2*T(3. 2) - 2*T(3, 3) = 9 + 12 - 2 = 19.
Z-sequence recurrence: 5 = T(4, 0) = 2*T(3, 0) - 1*T(3, 1) + 2*T(3, 2) - 6*T(3, 3) = 8 - 9 + 12 + 6 = 5.
Boas-Buck recurrence: 19 = T(4, 2) = (1/2)*((2 + 0)*T(2, 2) + (2 + 2*2)*T(3, 2)) = (1/2)*(2 + 36) = 19. (End)

Reinhard Zumkeller, First 100 rows of triangle, flattened

Diagonal sums are A008937(n+1).
Cf. A048739 (row sums), A008288, A005900 (column 3), A014820 (column 4)
Cf. A081277, A142978 by antidiagonals, A119328, A110271 (matrix inverse).
Cf. A001100, A002464, A010673, A040000, A112478, A133872, A383857.

a104698 n k = a104698_tabl !! (n-1) !! (k-1)
a104698_row n = a104698_tabl !! (n-1)
a104698_tabl = [1] : [2,1] : f [1] [2,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) ([0] ++ us ++ [0]) $
          zipWith (+) ([1] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Jul 17 2015

A104698 := proc(n, k) add(binomial(k, j)*binomial(n-j+1, n-k-j), j=0..n-k) ; end proc:
seq(seq(A104698(n, k), k=0..n), n=0..15); # R. J. Mathar, Sep 04 2011
T := (n, k) -> binomial(n + 1, k + 1)*hypergeom([-k, k - n], [-n - 1], -1):
for n from 0 to 9 do seq(simplify(T(n, k)), k = 0..n) od;
T := proc(n, k) option remember; if k = 0 then n + 1 elif k = n then 1 else T(n-2, k-1) + T(n-1, k-1) + T(n-1, k) fi end: # Peter Luschny, May 13 2025

u[1, ] = 1; v[1, ] = 1;
u[n_, x_] := u[n, x] = x u[n-1, x] + v[n-1, x] + 1;
v[n_, x_] := v[n, x] = 2 x u[n-1, x] + v[n-1, x] + 1;
Table[CoefficientList[u[n, x], x], {n, 1, 11}] // Flatten (* Jean-François Alcover, Mar 10 2019, after Clark Kimberling *)

T(n,k)=sum(j=0,n-k,binomial(k,j)*binomial(n-j+1,k+1)) \\ Charles R Greathouse IV, Jan 16 2012

The triangle is extracted from the product A * B; A = [1; 1, 1; 1, 1, 1; ...], B = [1; 1, 1; 1, 3, 1; 1, 5, 5, 1; ...] both infinite lower triangular matrices (rest of the terms are zeros). The triangle of matrix B by rows = A008288, Delannoy numbers.
From Paul Barry, Jul 18 2005: (Start)
Riordan array (1/(1-x)^2, x(1+x)/(1-x)) = (1/(1-x), x)*(1/(1-x), x(1+x)/(1-x)).
T(n, k) = Sum_{j=0..n} Sum_{i=0..j-k} C(j-k, i)*C(k, i)*2^i.
T(n, k) = Sum_{j=0..k} Sum_{i=0..n-k-j} (n-k-j-i+1)*C(k, j)*C(k+i-1, i). (End)
T(n, k) = binomial(n+1, k+1)*2F1([-k, k-n], [-n-1], -1) where 2F1 is a Gaussian hypergeometric function. - R. J. Mathar, Sep 04 2011
T(n, k) = T(n-2, k-1) + T(n-1, k-1) + T(n-1, k) for 1 < k < n; T(n, 0) = n + 1; T(n, n) = 1. - Reinhard Zumkeller, Jul 17 2015
From Wolfdieter Lang, May 13 2025: (Start)
The Riordan triangle T = (1/(1 - x)^2, x*(1 + x)/(1 - x)) has the o.g.f. G(x, y) = 1/((1 - x)*(1 - x - y*x*(1+x))) for the row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k.
The o.g.f. for column k is G(k, x) = (1/(1 - x)^2)*(x*(1 + x)/(1 - x))^k, for k >= 0.
The o.g.f. for the diagonal m is D(m, x) = N(m, x)/(1 - x)^(m+1), with the numerator polynomial N(m, x) = Sum_{k=0..floor(m/2)} A034867(m, k)*x^(2*k) for m >= 0.
The row sums with o.g.f. R(x) = 1/((1 -x)*(1 - 2*x -x^2) give A048739.
The alternating row sums with o.g.f. 1/((1 - x)(1 + x^2)) give A133872.
The A-sequence for this Riordan triangle has o.g.f. A(x) = 1 + x + sqrt(1 + 6*x + x^2))/2 giving A112478(n). Hence T(n, k) = Sum_{j=0..n-k} A112478(j)*T(n-1, k-1+j), for n >= 1, k >= 1, T(n, k) = 0 for n < k, and T(0, 0) = 1.
The Z-sequence has o.g.f. (5 + x - sqrt(1 + 6*x + x^2))/2 = 3 + x - A(x) giving Z(n) = {2, -1, -A112478(n >= 2)}. Hence T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), for n >= 1. For A- and Z-sequences of Riordan triangles see a W. Lang link at A006232 with references.
The Boas-Buck sequences alpha and beta for the Riordan triangle T (see A046521 for the Aug 10 2017 comment and reference) are alpha(n) = A040000(n+1) = repeat{2} and beta(n) = A010673(n+1) = repeat{2,0}. Hence the recurrence for column T(n, k){n>=k}, with input T(k, k) = 1, for k >= 0, is T(n, k) = (1/(n-k)) * Sum{j=k..n-1} (2 + k*(1 + (-1)^(n-1-j))) *T(j,k), for n >= k+1. (End)

A031138 Numbers k such that 1^5 + 2^5 + ... + k^5 is a square.

Original entry on oeis.org

1, 13, 133, 1321, 13081, 129493, 1281853, 12689041, 125608561, 1243396573, 12308357173, 121840175161, 1206093394441, 11939093769253, 118184844298093, 1169909349211681, 11580908647818721, 114639177128975533, 1134810862641936613, 11233469449290390601

Offset: 1

Ignacio Larrosa Cañestro, entry revised Feb 27 2000

Partial sums of A004291 or convolution of A040000 with A054320. - R. J. Mathar, Oct 26 2009
This is a 6th-degree Diophantine equation 12*m^2 = n^2*(n+1)^2*(2*n^2 + 2*n - 1) which reduces to the generalized Pell equation 6*q^2 = (2*n + 1)^2 - 3 where q = 3*m/(n*(n+1)), so there is no surprise that the solutions satisfy a linear recurrent equation. - Charles R Greathouse IV, Max Alekseyev, Oct 22 2012
Also k such that k^2 + (k+1)^2 is equal to the sum of three consecutive squares, for example 13^2 + 14^2 = 10^2 + 11^2 + 12^2. - Colin Barker, Sep 06 2015

			a(2) = 13 because 1^5+2^5+...13^5 = 1001^2; a(1) = 1 because 1^5 = 1^2.

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hex Number
Index entries for linear recurrences with constant coefficients, signature (11, -11, 1).

Cf. A000539, A006061, A054320, A007667.

[Round(-1/2 + ((3 - Sqrt(6))/4)*(5 + 2*Sqrt(6))^n + ((3 + Sqrt(6) )/4)*(5 - 2*Sqrt(6))^n): n in [0..50]]; // G. C. Greubel, Nov 04 2017

LinearRecurrence[{11,-11,1},{1,13,133},20 ] (* Harvey P. Dale, Oct 23 2012 *)

isok(n) = issquare(sum(i=1, n, i^5)); \\ Michel Marcus, Dec 28 2013

Vec(x*(1+x)^2/((1-x)*(x^2-10*x+1)) + O(x^40)) \\ Colin Barker, Sep 06 2015

a(n) = 11*(a(n-1) - a(n-2)) + a(n-3).
a(n) = -1/2 + ((3 - sqrt(6))/4)*(5 + 2*sqrt(6))^n + ((3 + sqrt(6))/4)*(5 - 2*sqrt(6))^n.
a(n)^2 + (a(n) + 1)^2 = (b(n) - 1)^2 + b(n)^2 + (b(n) + 1)^2 = c(n) = 3*d(n) + 2; where b(n) is A054320, c(n) is A007667 and d(n) is A006061.
a(n) = 10*a(n-1) - a(n-2) + 4; a(0) = a(1) = 1. Also sum of first a(n) fifth powers is a square m^2, where m has factors A000217{a(n)} and A054320(n). - Lekraj Beedassy, Jul 08 2002
contfrac(sqrt(6)/A054320(n))[4]/2 - Thomas Baruchel, Dec 02 2003
G.f.: x*(1+x)^2/((1-x)*(x^2-10*x+1)). - R. J. Mathar, Oct 26 2009

A176591 Bernoulli denominators A141056(n), with the exception a(1)=1.

Original entry on oeis.org

1, 1, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2, 64722, 2, 30, 2, 4686, 2, 140100870, 2, 6, 2, 30, 2

Offset: 0

Paul Curtz, Apr 21 2010

These are also the denominators of a sequence generated by inverse binomial transform of a modified Bernoulli sequence described in (with numerators in) A176328.

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A141056, A160014.

read("transforms") ; evb := [1, 0, seq(bernoulli(n), n=2..50)] ; BINOMIALi(evb) ; apply(denom, %) ; # R. J. Mathar, Dec 01 2010
seq(denom((bernoulli(i,1)+bernoulli(i,2))/2),i=0..50); # Peter Luschny, Jun 17 2012

a[n_] := If[OddQ[n], 2, BernoulliB[n] // Denominator]; a[1] = 1; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 29 2012 *)
Join[{1,1},BernoulliB[Range[2,80]]/.(0->1/2)//Denominator] (* Harvey P. Dale, Dec 31 2018 *)

A176591(n) = { my(p=1); if(n>1, fordiv(n, d, my(r=d+1); if(isprime(r), p = p*r))); return(p); }; \\ Antti Karttunen, Dec 20 2018, after code in A141056

a(n) = A141056(n), n <> 1.
a(n) = A027760(n), n>1.
a(2n) = A002445(n), a(2n+1)= A040000(n).

More terms from Antti Karttunen, Dec 20 2018

A383735 Array read by antidiagonals, where each row is the cluster series for percolation on the cells of a certain type of polyominoids.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 0, 1, 0, 2, 0, 2, 1, 0, 2, 0, 2, 4, 1, 0, 2, 0, 2, 12, 6, 1, 0, 2, 0, 2, 24, 18, 0, 1, 0, 2, 0, 2, 52, 48, 0, 4, 1, 0, 2, 0, 2, 108, 126, 0, 12, 4, 1, 0, 2, 0, 2, 224, 300, 0, 24, 12, 8, 1, 0, 2, 0, 2, 412, 762, 0, 52, 24, 32, 0, 1

Offset: 1

Pontus von Brömssen, May 10 2025

T(n,k) is the coefficient of p^(k+1), k >= 0, in the power series expansion of the expected finite size of the cluster containing a given cell for percolation with probability p on the polyominoid cells corresponding to row n of A366766. If the given cell is not open, its cluster is empty. Equivalently, T(n,k) can be taken to be the coefficient of p^k if we condition on the event that the given cell is open.

See A366766 for details on how the polyominoids are specified and on the ordering of the rows.

			Array begins:
  n\k| 0  1  2   3   4    5    6    7     8     9     10     11      12
  ---+-----------------------------------------------------------------
   1 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   2 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   3 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   4 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   5 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   6 | 1  6 18  48 126  300  762 1668  4216  8668  21988  43058  110832
   7 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   8 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   9 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
  10 | 1  8 32 108 348 1068 3180 9216 26452 73708 206872 563200 1555460
  11 | 1  0  0   0   0    0    0    0     0     0      0      0       0
  12 | 1  2  2   2   2    2    2    2     2     2      2      2       2

Index entries for sequences related to cluster series.

Cf. A366766, A366767, A366768.
Rows include:
   n | sequence for row n
  ---+-------------------
| A000007
| A040000
| A000007
| A040000
| A003203
| A003198
| A000007
| A003203
| A003203
| A003201
| A000007
| A040000
| A383737
| A003207
| A000007
| A003203
| A383737
| A383736
| A003203
| A003201
  ...
| A000007
| A003211
| A003209
| A036396
| A003210
  ...
| A036402
| A000007
| A040000
  ...
| A000007
| A003203
  ...
| A003203
| A003201

T(n,k) = [p^k] Sum_P m^2*p^(m-1)*(1-p)^j / binomial(D,d) = Sum_P m^2*(-1)^(k-m+1)*binomial(j,k-m+1) / binomial(D,d), where the sum is over all fixed polyominoids P (corresponding to row n of A366766), m is the number of cells of P, and j is the number of cells that are not in P but are adjacent to a cell in P; d is the dimension of the cells and D is the dimension of the ambient space. It is sufficient to take the sums over those P that have at most k+1 cells.

A089068 a(n) = a(n-1)+a(n-2)+a(n-3)+2 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 3, 6, 12, 23, 43, 80, 148, 273, 503, 926, 1704, 3135, 5767, 10608, 19512, 35889, 66011, 121414, 223316, 410743, 755475, 1389536, 2555756, 4700769, 8646063, 15902590, 29249424, 53798079, 98950095, 181997600, 334745776, 615693473

Offset: 0

Roger L. Bagula, Dec 03 2003

The a(n+2) represent the Kn12 and Kn22 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums. - Johannes W. Meijer, Sep 21 2010

Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Cf. A000931, A000073, A077939, A113300, A001057, A006054, A033505.

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25). - Johannes W. Meijer, Sep 21 2010

Join[{a=0,b=0,c=1},Table[d=a+b+c+2;a=b;b=c;c=d,{n,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+2}, a[n],{n,40}] (* or *) LinearRecurrence[{2,0,0,-1},{0,0,1,3},40] (* Harvey P. Dale, Sep 19 2011 *)

a(n) = A008937(n-2)+A008937(n-1). - Johannes W. Meijer, Sep 21 2010
a(n) = A018921(n-5)+A018921(n-4), n>4. - Johannes W. Meijer, Sep 21 2010
a(n) = A000073(n+2)-1. - R. J. Mathar, Sep 22 2010
From Johannes W. Meijer, Sep 22 2010: (Start)
a(n) = a(n-1)+A001590(n+1).
a(n) = Sum_{m=0..n} A040000(m)*A000073(n-m).
a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+1,k+1).
G.f. = x^2*(1+x)/((1-x)*(1-x-x^2-x^3)). (End)
a(n) = 2*a(n-1)-a(n-4), a(0)=0, a(1)=0, a(2)=1, a(3)=3. - Bruno Berselli, Sep 23 2010

Corrected and information added by Johannes W. Meijer, Sep 22 2010, Oct 22 2010

Definition based on arbitrarily set floating-point precision removed by R. J. Mathar, Sep 30 2010

A112739 Array counting nodes in rooted trees of height n in which the root and internal nodes have valency k (and the leaf nodes have valency one).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 5, 2, 1, 1, 5, 10, 7, 2, 1, 1, 6, 17, 22, 9, 2, 1, 1, 7, 26, 53, 46, 11, 2, 1, 1, 8, 37, 106, 161, 94, 13, 2, 1, 1, 9, 50, 187, 426, 485, 190, 15, 2, 1, 1, 10, 65, 302, 937, 1706, 1457, 382, 17, 2, 1, 1, 11, 82, 457, 1814, 4687, 6826, 4373, 766, 19

Offset: 0

Paul Barry, Sep 16 2005

Rows of the square array have g.f. (1+x)/((1-x)(1-kx)). They are the partial sums of the coordination sequences for the infinite tree of valency k. Row sums are A112740.

Rows of the square array are successively: A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023. - Philippe Deléham, Feb 22 2014

			As a square array, rows begin
1,1,1,1,1,1,... (A000012)
1,2,2,2,2,2,... (A040000)
1,3,5,7,9,11,... (A005408)
1,4,10,22,46,94,... (A033484)
1,5,17,53,161,485,... (A048473)
1,6,26,106,426,1706,... (A020989)
1,7,37,187,937,4687,... (A057651)
1,8,50,302,1814,10886,... (A061801)
As a number triangle, rows start
1;
1,1;
1,2,1;
1,3,2,1;
1,4,5,2,1;
1,5,10,7,2,1;

L. He, X. Liu and G. Strang, (2003) Trees with Cantor Eigenvalue Distribution. Studies in Applied Mathematics 110 (2), 123-138.
L. He, X. Liu and G. Strang, Laplacian eigenvalues of growing trees, Proc. Conf. on Math. Theory of Networks and Systems, Perpignan (2000).

Cf. A112468, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023.

As a square array read by antidiagonals, T(n, k)=sum{j=0..k, (2-0^j)*(n-1)^(k-j)}; T(n, k)=(n(n-1)^k-2)/(n-2), n<>2, T(2, n)=2n+1; T(n, k)=sum{j=0..k, (n(n-1)^j-0^j)/(n-1)}, j<>1. As a triangle read by rows, T(n, k)=if(k<=n, sum{j=0..k, (2-0^j)*(n-k-1)^(k-j)}, 0).

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A040012 Continued fraction for sqrt(17).

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A212119 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the number of divisors d of n with min(d, n/d) = k.

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A225094 Number A(n,k) of lattice paths without interior points from {n}^k to {0}^k using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A080934 Square array read by antidiagonals of number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2n steps with all values less than or equal to k.

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A104698 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k, j)*binomial(n-j+1, k+1).

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A031138 Numbers k such that 1^5 + 2^5 + ... + k^5 is a square.

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A176591 Bernoulli denominators A141056(n), with the exception a(1)=1.

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