A047891 -id:A047891 - cp's OEIS frontend

A372214 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of G(x)^n evaluated at x = 1, where G(x) = (1 - 2x - sqrt(1 - 8x + 4x^2))/(2x).

1, 4, 40, 487, 6376, 86629, 1203823, 16984678, 242274280, 3484593028, 50444222665, 734066291974, 10728052396111, 157349171819155, 2314894133906086, 34145661019248487, 504810905195542504, 7478066502444399439, 110972913533524676080, 1649407167353221551706, 24549982881130265421001

Offset: 0

Peter Bala, Apr 23 2024

x*G(x) = (1 - 2*x - sqrt(1 - 8*x + 4*x^2))/2 is the o.g.f. of A047891.
The sequence satisfies the Gauss congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all primes 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 all primes p >= 5 and positive integers n and k. Examples of these supercongruences are given below.
More generally, for each integer m, we conjecture that the sequence {a_m(n) : n >= 0}, defined by setting a_m(n) = the n-th order Taylor polynomial of G(x)^(m*n) evaluated at x = 1, satisfies the same supercongruences.

			n-th order Taylor polynomial of G(x)^n:
  n = 0: G(x)^0 = 1 + O(x)
  n = 1: G(x)^1 = 1 + 3*x + O(x^2)
  n = 2: G(x)^2 = 1 + 6*x + 33*x^2 + O(x^3)
  n = 3: G(x)^3 = 1 + 9*x + 63*x^2 + 414*x^3 + O(x^4)
  n = 4: G(x)^4 = 1 + 12*x + 102*x^2 + 768*x^3 + 5493*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 3 = 4, a(2) = 1 + 6 + 33 = 40, a(3) = 1 + 9 + 63 + 414 = 487 and a(4) = 1 + 12 + 102 + 768 + 5493 = 6376.
The triangle of coefficients of the n-th order Taylor polynomial of G(x)^n, n >= 0, in descending powers of x begins
                                             row sums
  n = 0 |    1                                   1
  n = 1 |    3    1                              4
  n = 2 |   33    6     1                       40
  n = 3 |  414   63     9    1                 487
  n = 4 | 5493  768   102   12   1            6376
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group.
Examples of supercongruences:
a(13) - a(1) = 157349171819155 - 4 = (3^3)*(13^3)*269*9860941 == 0 (mod 13^3).
a(2*7) - a(2) = 2314894133906086 - 40 = 2*(3^4)*(7^3)*11*12119*312509 == 0 (mod 7^3).

Cf. A047891, A219535, A333090, A333093, A372215.

G := x -> (1/2)*(1 - 2*x - sqrt(1 - 8*x + 4*x^2))/x:
H := (x, n) -> series(G(x)^n, x, 41):
seq(add(coeff(H(x, n), x, k), k = 0..n), n = 0..20);

Table[SeriesCoefficient[(2*(1 + x)^2/(1 - x + Sqrt[1 - 6*x - 3*x^2]))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 02 2024 *)

G(x) = (1 - 2*x - sqrt(1 - 8*x + 4*x^2))/(2*x);
a(n) = my(x='x+O('x^(n+2))); subst(Pol(Vec(G(x)^n)), 'x, 1); \\ Michel Marcus, May 07 2024

a(n) = [x^n] ( (1 + x)*G(x/(1 + x)) )^n.
O.g.f.: ( 1 + x*F'(x)/F(x) )/( 1 - x*F(x) ), where F(x) = (1/x)*Revert( x/G(x) ) = = 1 + 3*x + 21*x^2 + 192*x^3 + 2001*x^4 + ... is the o.g.f. of A219535.
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) ~ sqrt(1 + 17/sqrt(33)) * (59 + 11*sqrt(33))^n / (sqrt(3*Pi*n) * 2^(3*n + 3/2)). - Vaclav Kotesovec, May 02 2024

A357585 Triangle read by rows. Inverse of the convolution triangle of A108524, the number of ordered rooted trees with n generators.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 7, 4, 1, 0, 32, 18, 6, 1, 0, 166, 92, 33, 8, 1, 0, 926, 509, 188, 52, 10, 1, 0, 5419, 2964, 1113, 328, 75, 12, 1, 0, 32816, 17890, 6792, 2078, 520, 102, 14, 1, 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1

Offset: 0

Peter Luschny, Oct 08 2022

Also the matrix inverse of the signed version of A105475 with 1, 0, 0, 0, ... as column 0.

			Triangle T(n, k) starts:
[0] 1;
[1] 0,      1;
[2] 0,      2,      1;
[3] 0,      7,      4,     1;
[4] 0,     32,     18,     6,     1;
[5] 0,    166,     92,    33,     8,    1;
[6] 0,    926,    509,   188,    52,   10,  1;
[7] 0,   5419,   2964,  1113,   328,   75,  12,   1;
[8] 0,  32816,  17890,  6792,  2078,  520, 102,  14,  1;
[9] 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1;

Cf. A108524 (column 1), A047891 (row sums), A105475.

InvPMatrix := proc(dim, seqfun) local k, m, M, A;
    if dim < 1 then return [] fi;
    A := [seq(seqfun(i), i = 1..dim-1)];
    M := Matrix(dim, shape=triangular[lower]); M[1, 1] := 1;
    for m from 2 to dim do
        M[m, m] := M[m - 1, m - 1] / A[1];
        for k from m-1 by -1 to 2 do
            M[m, k] := M[m - 1, k - 1] -
                add(A[i+1] * M[m, k + i], i = 1..m-k) / A[1]
od od; M end:
InvPMatrix(10, n -> [1, -2][irem(n-1, 2) + 1]);

A379103 Expansion of (1-3x-sqrt(9x^2-14*x+1))/4.

Original entry on oeis.org

0, 1, 5, 35, 295, 2765, 27705, 290535, 3148995, 34995065, 396602605, 4566227435, 53259218495, 627982592965, 7473163652705, 89640387354735, 1082664905352795, 13155505626756465, 160709002086562005, 1972595405313408435, 24315686632846439895, 300886761671728853565, 3736205372071338170505, 46540791299676591116535

Offset: 0

Nathaniel Johnston, Dec 15 2024

Problem A6 on the 2024 William Lowell Putnam Mathematical Competition was to compute the Hankel transform of this sequence, which is A110147.

Given constants X and Y, let A(x) = (1 - x*(X - Y) - sqrt(1 - 2*x*(X + Y) + x^2*(X - Y)^2))/(2*Y) = x*(1) + x^2*(X) + x^3*X*(X + Y) + x^4*X*(X^2 + 3*X*Y + Y^2) + ... where the coefficients of A(x) is the Narayana triangle A090181. A(x) satisfies 0 = x - A(x)*(1 - x*(X-Y)) + A(x)^2*Y. The Hankel transform of the coefficients 1, X, X*(X + Y), ... is the sequence 1, (X*Y), (X*Y)^2, ... while the Hankel transform of X, X*(X + Y), X*(X^2 + 3*X*Y + Y^2), ... is the sequence X, X^3*Y, X^6*Y^3, X^10*Y^6, .... In the case of this sequence, X = 5 and Y = 2. - Michael Somos, Apr 26 2025

			G.f. = x + 5*x^2 + 35*x^3 + 295*x^4 + 2765*x^5 + 27705*x^6 + ... - _Michael Somos_, Apr 26 2025

Nathaniel Johnston, Determinant involving (1 - 3x - sqrt(1 - 14x + 9x^2))/4, YouTube video, 2024.
Nathaniel Johnston, Putnam 2024 A6 solution, 2024.

Cf. A025231, A047891, A082298, A110147.

a = 3;b = 2;c(1) = 1;last_val = 16;for j = 2:last_val
c(j) = a*c(j-1) + b*sum(c(1:j-1).*fliplr(c(1:j-1)));
end

a[ n_] := SeriesCoefficient[ (1 - 3*x - Sqrt[1 - 14*x + 9*x^2])/4, {x, 0, n}]; (* Michael Somos, Apr 26 2025 *)
a[ n_] := With[{X = 5, Y = 2}, SeriesCoefficient[ Nest[x/(1 - (X-Y)*x - Y*#)&, O[x], n], {x, 0, n}]]; (* Michael Somos, Apr 28 2025 *)
a[ n_] := With[{X = 5, Y = 2}, SeriesCoefficient[ Nest[x/(1 - X*x/(1 - Y*#))&, O[x], Ceiling[n/2]], {x, 0, n}]]; (* Michael Somos, Apr 28 2025 *)

my(x='x+O('x^33)); concat([0],Vec((1-3*x-sqrt(9*x^2-14*x+1))/4)) \\ Joerg Arndt, Dec 15 2024

a(n) = my(A = O(x)); for(k=1, n, A = x + 3*x*A + 2*A^2); polcoeff(A, n); /* Michael Somos, Apr 26 2025 */

a(n) = my(A = O(x), X = 5, Y = 2); for(k = 1, n, A = x/(1 - (X-Y)*x - Y*A)); polcoeff(A, n); /* Michael Somos, Apr 28 2025 */

a(n) = my(A = O(x), X = 5, Y = 2); for(k = 1, (n+1)\2, A = x/(1 - X*x/(1 - Y*A))); polcoeff(A, n); /* Michael Somos, Apr 28 2025 */

a(0) = 0, a(1) = 1, a(n) = 3*a(n-1) + 2*Sum_{k=0..n} a(k)*a(n-k) for n >= 2.
G.f.: (1-3*x-sqrt(9*x^2-14*x+1))/4.
G.f.: x/(1-5*x/(1-2*x/(1-5*x/(1-2*x/(1-5*x/(...)))))). - Thomas Scheuerle, Feb 28 2025
a(n) = (1/4)*(-1)^(n+1) * Sum_{k=0..n} binomial(1/2,k) * binomial(1/2,n-k) * (7+2*sqrt(10))^k * (7-2*sqrt(10))^(n-k) for n >= 2. - Ehren Metcalfe, Feb 26 2025
a(n) ~ 5^(1/4) * (7 + 2*sqrt(10))^(n - 1/2) / (2^(7/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 27 2025
The g.f. A(x) satisfies 0 = x - (1 - 3*x)*A(x) + 2*A(x)^2 and A(x) = x + 3*x*A(x) + 2*A(x)^2. - Michael Somos, Apr 26 2025

A247507 Square array read by ascending antidiagonals, n>=0, k>=0. Row n is the expansion of (1-nx-sqrt(n^2x^2-2nx-4x+1))/(2x).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 22, 14, 1, 5, 20, 57, 90, 42, 1, 6, 30, 116, 300, 394, 132, 1, 7, 42, 205, 740, 1686, 1806, 429, 1, 8, 56, 330, 1530, 5028, 9912, 8558, 1430, 1, 9, 72, 497, 2814, 12130, 35700, 60213, 41586, 4862

Offset: 0

Peter Luschny, Nov 17 2014

			   [0][1] [2]  [3]    [4]     [5]      [6]       [7]
[0] 1, 1,  2,   5,    14,     42,     132,      429,.. A000108
[1] 1, 2,  6,  22,    90,    394,    1806,     8558,.. A006318
[2] 1, 3, 12,  57,   300,   1686,    9912,    60213,.. A047891
[3] 1, 4, 20, 116,   740,   5028,   35700,   261780,.. A082298
[4] 1, 5, 30, 205,  1530,  12130,  100380,   857405,.. A082301
[5] 1, 6, 42, 330,  2814,  25422,  239442,  2326434,.. A082302
[6] 1, 7, 56, 497,  4760,  48174,  507696,  5516133,.. A082305
[7] 1, 8, 72, 712,  7560,  84616,  985032, 11814728,.. A082366
[8] 1, 9, 90, 981, 11430, 140058, 1782900, 23369805,.. A082367

L. Yang, S.-L. Yang, A relation between Schroder paths and Motzkin paths, Graphs Combinat. 36 (2020) 1489-1502, eq. (5).

Cf. A243631.

Main diagonal gives A302286.

gf := n -> (1-n*x-sqrt(n^2*x^2-2*n*x-4*x+1))/(2*x):
for n from 0 to 10 do lprint(PolynomialTools:-CoefficientList( convert(series(gf(n),x,8),polynom),x)) od;

G.f. of row n: 1/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 06 2018

Offset changed to 0 by Alois P. Heinz, May 28 2015

A346505 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^2) / (1 - x + 2 * x^2).

Original entry on oeis.org

1, 2, 4, 12, 44, 172, 700, 2940, 12652, 55500, 247260, 1115740, 5088908, 23423020, 108659324, 507520316, 2384733868, 11264884876, 53464215580, 254822253852, 1219182031820, 5853309920748, 28190437248700, 136160853462524, 659401832797676, 3201141695492172, 15575294057678428

Offset: 0

Ilya Gutkovskiy, Jul 21 2021

nonn

Cf. A004148, A006318, A047891, A078482, A082582, A346506.

nmax = 26; A[] = 0; Do[A[x] = (1 + x A[x]^2)/(1 - x + 2 x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[a[k] a[n - k - 1], {k, 2, n - 1}]; Table[a[n], {n, 0, 26}]
CoefficientList[Series[(1 - x*(1 - 2*x)) * (1 - Sqrt[1 - 4*x/(-1 + x - 2*x^2)^2]) / (2*x), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 26 2023 *)

a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=2..n-1} a(k) * a(n-k-1).
From Vaclav Kotesovec, Sep 26 2023: (Start)
G.f.: (1 - x*(1 - 2*x))*(1 - sqrt(1 - 4*x/(-1 + x - 2*x^2)^2))/(2*x).
a(n) ~ sqrt((69 + 57*sqrt(114) + 23*3^(5/6)*sqrt(38)*(9 + 2*sqrt(114))^(1/3) - 36*3^(1/6)*sqrt(38)*(9 + 2*sqrt(114))^(2/3) + 291*(27 + 6*sqrt(114))^(1/3) - 54*(27 + 6*sqrt(114))^(2/3))/(-72 - 16*sqrt(114) + 3^(11/6)*sqrt(38)*(9 + 2*sqrt(114))^(1/3) + 3^(1/6)*sqrt(38)*(9 + 2*sqrt(114))^(2/3) + 26*(27 + 6*sqrt(114))^(1/3) - 6*(27 + 6*sqrt(114))^(2/3))) * 2^(n - 1/2) * 3^(1/6 + 4*n/3) * ((9 + 2*sqrt(114))^((1/3)*(n-1)) / (sqrt(Pi) * n^(3/2) * (-15 + (27 + 6*sqrt(114))^(2/3))^n)). (End)

A349012 G.f. A(x) satisfies: A(x) = (1 + x * A(2x)) / (1 - x A(x)).

Original entry on oeis.org

1, 2, 8, 52, 552, 10208, 350112, 23159760, 3012389984, 777296223040, 399542726439936, 409933997609848192, 840366306053838941952, 3443822768422065940362240, 28218687132517064788995222528, 462391421142204650963524251763968, 15152566983759983965941543133445666304

Offset: 0

Ilya Gutkovskiy, Nov 05 2021

nonn

Cf. A006318, A047891, A135867, A348875, A349013.

nmax = 16; A[] = 0; Do[A[x] = (1 + x A[2 x])/(1 - x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 2^(n - 1) a[n - 1] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

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

a(n) ~ c * 2^(n*(n-1)/2), where c = 11.40022022373995418911523299051117421707893086825818379118899572625286143... - Vaclav Kotesovec, Nov 06 2021

A371392 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+2*x)^3 ).

Original entry on oeis.org

1, 7, 68, 769, 9492, 124014, 1686120, 23610565, 338200148, 4932348226, 72993007672, 1093371638954, 16545598769416, 252567107648604, 3884497559034192, 60136704175071789, 936373570430169300, 14654788984834217850, 230405413840884827160, 3637362857723455772670

Offset: 0

Seiichi Manyama, Mar 21 2024

nonn

Index entries for reversions of series

Cf. A047891, A371391.

Cf. A113207.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+2*x)^3)/x)

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

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

A133366 Triangle T(n,k)read by rows given by [3,1,3,1,3,1,3,1,3,1,3,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 3, 1, 12, 7, 1, 57, 43, 11, 1, 300, 262, 90, 15, 1, 1686, 1618, 667, 153, 19, 1, 9912, 10159, 4745, 1336, 232, 23, 1, 60213, 64783, 33147, 10785, 2333, 327, 27, 1, 374988, 418786, 229726, 83286, 21098, 3722, 438, 31, 1

Offset: 0

Philippe Deléham, Oct 27 2007

A121576*A007318 as infinite lower triangular matrices.

			Triangle begins:
       1;
       3,      1;
      12,      7,      1;
      57,     43,     11,     1;
     300,    262,     90,    15,     1;
    1686,   1618,    667,   153,    19,    1;
    9912,  10159,   4745,  1336,   232,   23,   1;
   60213,  64783,  33147, 10785,  2333,  327,  27,  1;
  374988, 418786, 229786, 83286, 21098, 3722, 438, 31, 1; ...

Cf. A047891, A000012, A004767.

Cf. A007318, A121576.

T(0,0)=1; T(n,k) = 0 if k < 0 or if k > n; T(n,0) = 3*T(n-1,0) + 3*T(n-1,1); T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + 3*T(n-1,k+1) for k >= 1.

Sum_{k>=0} T(m,k)*T(n,k)*3^k = T(m+n,0)= A047891(m+n+1). - Philippe Deléham, Jan 24 2010

cp's OEIS Frontend

A372214 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of G(x)^n evaluated at x = 1, where G(x) = (1 - 2*x - sqrt(1 - 8*x + 4*x^2))/(2*x).

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A357585 Triangle read by rows. Inverse of the convolution triangle of A108524, the number of ordered rooted trees with n generators.

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A379103 Expansion of (1-3*x-sqrt(9*x^2-14*x+1))/4.

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A247507 Square array read by ascending antidiagonals, n>=0, k>=0. Row n is the expansion of (1-n*x-sqrt(n^2*x^2-2*n*x-4*x+1))/(2*x).

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A346505 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^2) / (1 - x + 2 * x^2).

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A349012 G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x * A(x)).

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A371392 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+2*x)^3 ).

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A133366 Triangle T(n,k)read by rows given by [3,1,3,1,3,1,3,1,3,1,3,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A372214 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of G(x)^n evaluated at x = 1, where G(x) = (1 - 2x - sqrt(1 - 8x + 4x^2))/(2x).

A379103 Expansion of (1-3x-sqrt(9x^2-14*x+1))/4.

A247507 Square array read by ascending antidiagonals, n>=0, k>=0. Row n is the expansion of (1-nx-sqrt(n^2x^2-2nx-4x+1))/(2x).

A349012 G.f. A(x) satisfies: A(x) = (1 + x * A(2x)) / (1 - x A(x)).