A333090 -id:A333090 - cp's OEIS frontend

A333092 a(n) is the n-th order Taylor polynomial (centered at 0) of S(x)^(3n) evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2*x) is the o.g.f. of the Schröder numbers A006318.

1, 7, 109, 1951, 36993, 724007, 14457421, 292732671, 5987886081, 123440423047, 2560421160109, 53373725431583, 1117198199782785, 23465732683090471, 494330214846965389, 10440064992542621951, 220978578227187097601, 4686426367646858888711, 99559270036968523118317

Offset: 0

Peter Bala, Mar 22 2020

The sequence satisfies the Gauss congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k.
We conjecture that the sequence satisfies the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. Examples of these congruences are given below.
More generally, for each integer m, we conjecture that the sequence a_m(n) defined as the n-th order Taylor polynomial of S(x)^(m*n) evaluated at x = 1 satisfies the same congruences. For cases, see A333090 (m = 1) and A333091 (m = 2). For similarly defined sequences see A333093 through A333097.

			n-th order Taylor polynomial of S(x)^(3*n):
  n = 0: S(x)^0 = 1 + O(x)
  n = 1: S(x)^3 = 1 + 6*x + O(x^2)
  n = 2: S(x)^6 = 1 + 12*x + 96*x^2 + O(x^3)
  n = 3: S(x)^9 = 1 + 18*x + 198*x^2 + 1734*x^3 + O(x^4)
  n = 4: S(x)^12 = 1 + 24*x + 336*x^2 + 3608*x^3 + 33024*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 6 = 7, a(2) = 1 + 12 + 96 = 109, a(3) = 1 + 18 + 198 + 1734 = 1951 and a(4) = 1 + 24 + 336 + 3608 + 33024 = 36993.
The triangle of coefficients of the n-th order Taylor polynomial of S(x)^(2*n), n >= 0, in descending powers of x begins
                                           row sums
  n = 0 |     1                                1
  n = 1 |     6    1                           7
  n = 2 |    96   12    1                    109
  n = 3 |  1734  198   18   1               1951
  n = 4 | 33024 3608  336  24   1          36993
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group. The first column sequence [1, 6, 96, 1734, 33024, 648006, ...]  = [x^n] S(x)^(3*n), and may also satisfy the above congruences.
Examples of congruences:
a(13) - a(1) = 23465732683090471 - 7 = (2^5)*(3^4)*(13^3)*83*911*54497 == 0 ( mod 13^3 ).
a(3*7) - a(3) = 962815680123979633351467303 - 1951 = (2^3)*(7^3)*29*41* 1832861*161008076794727  == 0 ( mod 7^3 ).
a(5^2) - a(5) = 201479167004032422703424646224007 - 724007 = (2^5)*(5^6)* 402958334008064845406849291 == 0 ( mod 5^6 ).

Cf. A006318, A333090 through A333097.

S:= x -> (1/2)*(1-x-sqrt(1-6*x+x^2))/x:
G := (x,n) -> series(S(x)^(3*n), x, 101):
seq(add(coeff(G(x, n), x, n-k), k = 0..n), n = 0..25);

Table[SeriesCoefficient[((1+x)*(1 - 3*x*(1+x) + (x^2 + x - 1)*Sqrt[1 - 4*x*(1+x)]) / (2*x^3))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = [x^n] ( (1 + x)*S^3(x/(1 + x)) )^n.
O.g.f.: ( 1 + x*f'(x)/f(x) )/( 1 - x*f(x) ), where f(x) = 1 + 6*x + 66*x^2 + 902*x^3 + 13794*x^4 + ... = (1/x) * series reversion of ( x/S^3(x) ).
Row sums of the Riordan array ( 1 + x*f'(x)/f(x), x*f(x) ) belonging to the Hitting time subgroup of the Riordan group.
a(n) ~ 3*sqrt(85 + 21*sqrt(17)) * (349 + 85*sqrt(17))^n / (68 * sqrt(Pi*n) * 2^(5*n)). - Vaclav Kotesovec, Mar 28 2020

A351859 a(n) = [x^n] (1 + x + x^2 + x^3)^(4n)/(1 + x + x^2)^(3n).

Original entry on oeis.org

1, 1, 3, 19, 67, 251, 1137, 4803, 20035, 87013, 377753, 1634469, 7134385, 31261114, 137121113, 603206144, 2660097603, 11749336328, 51981371895, 230336544210, 1021976441817, 4539784391763, 20188837618799, 89871081815631, 400427435522737, 1785639575031501

Offset: 0

Peter Bala, Mar 01 2022

This sequence is the third in an infinite family of sequences defined as follows. Let k be a positive integer. Define the rational function G_k(x) = (1 + x + ... + x^k)^(k+1)/(1 + x + ... + x^(k-1))^k, so that G_1(x) = (1 + x)^2, and define the sequence u_k by u_k(n) = [x^n] G_k(x)^n. See A000984, the sequence of central binomial coefficients, for the case k = 1 and A351858 for the case k = 2. The present sequence is the case k = 3.
Given a power series G(x) with integer coefficients it is known that the sequence (g(n))n>=1 defined by g(n) := [x^n] G(x)^n satisfies the Gauss congruences g(n*p^r) == g(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Thus a(n) satisfies the Gauss congruences. Calculation suggests that, in fact, the stronger supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for primes p >= 5 and positive integers n and r. These supercongruences are known to hold for the central binomial coefficients A000984(n) = [x^n] ((1 + x)^2)^n (Meštrović, equation 39).
More generally, if r is a positive integer and s an integer then the sequence defined by a(r,s;n) = [x^(r*n)] G_3(x)^(s*n) may satisfy the same supercongruences.

			Examples of supercongruences:
a(5) - a(1) = 251 - 1 = 2*(5^3) == 0 (mod 5^3)
a(2*7) - a(2) = 137121113 - 3 = 2*5*(7^4)*5711 == 0 (mod 7^4)
a(5^2) - a(5) = 1785639575031501 - 251 = 2*(5^6)*1373*3989*10433 == 0 (mod 5^6)

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Paolo Xausa, Table of n, a(n) for n = 0..425
R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2011.

Cf. A000984, A103885, A333090, A333564, A333565, A333579, A333715, A351856, A351857, A351858.

seq(add(add(add((-1)^j*binomial(4*n,n-2*i-j-k)*binomial(4*n,i)* binomial(3*n+j-1,j)*binomial(j,k), k = 0..j), j = 0..n), i = 0..n), n = 0..25);

A351859[n_] := Sum[(-1)^j*Binomial[4*n, n-2*i-j-k]*Binomial[4*n, i]*Binomial[3*n+j-1, j]*Binomial[j, k], {i, 0, n}, {j, 0, n}, {k, 0, j}];
Array[A351859, 25, 0] (* Paolo Xausa, May 30 2025 *)

a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,j,(-1)^j*binomial(4*n,n-2*i-j-k)*binomial(4*n,i)*binomial(3*n+j-1,j)*binomial(j,k))));
vector(25,n,a(n-1)) \\ Paolo Xausa, May 04 2022

a(n) = Sum_{i = 0..n} Sum_{j = 0..n} Sum_{k = 0..j} (-1)^j* C(4n,n-2*i-j-k) *C(4n,i)*C(3n+j-1,j)*C(j,k).
The o.g.f. A(x) = 1 + x + 3*x^2 + 19*x^3 + 67*x^4 + ... is the diagonal of the bivariate rational function 1/(1 - t*(1 + x + x^2 + x^3)^4/(1 + x + x^2)^3) and hence is an algebraic function over the field of rational functions Q(x) by Stanley, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*(1 + x + x^2)^3/(1 + x + x^2 + x^3)^4 ) = 1 + x + 2*x^2 + 8*x^3 + 25*x^4 + 81*x^5 + 305*x^6 + .... Then A(x) = 1 + x*F'(x)/F(x).

A367639 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^2 / (1 + x).

Original entry on oeis.org

1, 3, 6, 16, 52, 184, 688, 2672, 10672, 43552, 180800, 761088, 3241088, 13937408, 60435968, 263962880, 1160188672, 5127762432, 22775636992, 101608357888, 455105255424, 2045751037952, 9225923895296, 41731062358016, 189275050729472, 860630181167104

Offset: 0

Seiichi Manyama, Nov 25 2023

Cf. A006318, A025227, A367640, A333090, A367641.

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

G.f.: A(x) = 2*(1+x)^2 / (1+sqrt(1-4*x*(1+x))).
a(n) = Sum_{k=0..n} binomial(k+2,n-k) * binomial(2*k,k)/(k+1).
a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 25 2023
D-finite with recurrence (n+1)*a(n) +(-3*n+1)*a(n-1) +2*(-4*n+9)*a(n-2) +4*(-n+4)*a(n-3)=0. - R. J. Mathar, Dec 04 2023
From Peter Bala, May 05 2024: (Start)
A(x) = (1 + x)*S(x/(1 + x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the g.f. of the large Schröder numbers A006318. Cf. A025227.
A333090(n) = [x^n] A(x)^n. (End)

A336858 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1, k) + T(n-1,k-1) with T(n,0) = T(n, n) = 1 (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 9, 1, 1, 7, 21, 31, 1, 1, 9, 37, 89, 121, 1, 1, 11, 57, 183, 393, 515, 1, 1, 13, 81, 321, 897, 1805, 2321, 1, 1, 15, 109, 511, 1729, 4431, 8557, 10879, 1, 1, 17, 141, 761, 3001, 9161, 22149, 41585, 52465, 1, 1, 19, 177, 1079, 4841, 17003, 48313, 112047, 206097, 258563, 1

Offset: 0

Petros Hadjicostas, Aug 05 2020

This is J. M. Bergot's triangular array described in A104858 with the top vertex of the triangle shifted from (1,1) to (0,0).

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   9,   1;
  1,  7,  21,  31,    1;
  1,  9,  37,  89,  121,    1;
  1, 11,  57, 183,  393,  515,    1;
  1, 13,  81, 321,  897, 1805, 2321,     1;
  1, 15, 109, 511, 1729, 4431, 8557, 10879, 1;
  ...

Cf. A001003, A005408, A006318, A008288, A035011, A059993, A086616, A104858, A333090.

A336858row := proc(n) option remember; local T, k, row;
row := Array(0..n, fill=1);
if n = 0 then return row fi; T := procname(n-1);
for k from 1 to n-1 do row[k] := T[k] + T[k-1] + row[k-1] od; row end:
T := (n, k) -> A336858row(n)[k]:
seq(print(seq(T(n, k), k=0..n)), n=0..8); # Peter Luschny, Aug 06 2020

T[, 0] = 1; T[n, n_] = 1;
T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, k] + T[n-1, k-1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)

T(n,k) = T(n, k-1) + T(n-1, k) + T(n-1, k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.
T(n,k) = D(n,k) - Sum_{m=1..k} b(m-1)*D(n-m, k-m) - Sum_{m=0..k-1} D(n-m, k-m-1), where D(n,k) = A008288(n,k) (square array of Delannoy numbers) and b(n) = A086616(n).
T(n,1) = A005408(n-1) = 2*n - 1 for n >= 1.
T(n,2) = A059993(n-2) = 2*n^2 - 2*n - 3 for n >= 2.
T(n,n-1) = A086616(n-1) for n >= 1.
T(n,n-2) = A035011(n-1) = A006318(n-1) - 1 for n >= 2.
Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.
Bivariate o.g.f.: (1 - y - x*y*(1 + g(x*y)))/((1 - x*y)*(1 - x - y - x*y)), where g(w) = 2/(1 - w + sqrt(1 - 6*w + w^2)) = o.g.f. of A006318 (large Schroeder numbers).
Bivariate o.g.f.: (1 - y - 2*x*y*q(x*y))/((1 - x*y)*(1 - x - y - x*y)), where q(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = o.g.f. of A001003 (little Schroeder numbers).
T(2*n,n) = A333090(n). - Peter Luschny, Aug 06 2020

A336859 Mirror image of triangular array A336858.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 5, 1, 1, 31, 21, 7, 1, 1, 121, 89, 37, 9, 1, 1, 515, 393, 183, 57, 11, 1, 1, 2321, 1805, 897, 321, 81, 13, 1, 1, 10879, 8557, 4431, 1729, 511, 109, 15, 1, 1, 52465, 41585, 22149, 9161, 3001, 761, 141, 17, 1, 1, 258563, 206097, 112047, 48313, 17003, 4841, 1079, 177, 19, 1

Offset: 0

Petros Hadjicostas, Aug 05 2020

This is a mirror image of A336858, which is a shifted version of J. M. Bergot's triangular array first described in A104858.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,     1;
  1,     3,    1;
  1,     9,    5,    1;
  1,    31,   21,    7,    1;
  1,   121,   89,   37,    9,   1;
  1,   515,  393,  183,   57,  11,   1;
  1,  2321, 1805,  897,  321,  81,  13,  1;
  1, 10879, 8557, 4431, 1729, 511, 109, 15, 1;
  ...

Cf. A001003, A006318, A086616, A104858, A333090, A336858.

A000108(n) = binomial(2*n, n)/(n+1);
A086616(n) = sum(k=0, n, binomial(n+k+1, 2*k+1) * A000108(k));
T(n, k) = if ((k==0) || (n==k), 1, if ((n<0) || (k<0), 0, if (k==1, A086616(n-1), if (n>k, T(n, k-1) - T(n-1, k-1) - T(n-1, k-2), 0))));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 08 2020

T(n,k) = A336858(n, n-k) for 0 <= k <= n.
T(n,k) = T(n, k-1) - T(n-1, k-1) - T(n-1, k-2) for 2 <= k <= n with T(n,0) = T(n,n) = 1 for n >= 0 and T(n,1) = A086616(n-1) for n >= 1.
T(2*n,n) = A333090(n).
Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.
Bivariate o.g.f.: (x*y*(1 + g(x)) + 1 - y)/((1 - x)*(1 - y + x*y + x*y^2)), where g(w) = 2/(1 - w + sqrt(1 - 6*w + w^2)) = o.g.f. of A006318 (large Schroeder numbers).
Bivariate o.g.f.: (2*x*y*q(x) + 1 - y)/((1 - x)*(1 - y + x*y + x*y^2)), where q(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = o.g.f. of A001003 (little Schroeder numbers).

cp's OEIS Frontend

A333092 a(n) is the n-th order Taylor polynomial (centered at 0) of S(x)^(3*n) evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the Schröder numbers A006318.

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A351859 a(n) = [x^n] (1 + x + x^2 + x^3)^(4*n)/(1 + x + x^2)^(3*n).

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A367639 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^2 / (1 + x).

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A336858 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1, k) + T(n-1,k-1) with T(n,0) = T(n, n) = 1 (n >= 0, 0 <= k <= n).

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A336859 Mirror image of triangular array A336858.

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A333092 a(n) is the n-th order Taylor polynomial (centered at 0) of S(x)^(3n) evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2*x) is the o.g.f. of the Schröder numbers A006318.

A351859 a(n) = [x^n] (1 + x + x^2 + x^3)^(4n)/(1 + x + x^2)^(3n).