A046714 -id:A046714 - cp's OEIS frontend

A046748 Row sums of triangle A046521.

1, 3, 13, 61, 295, 1447, 7151, 35491, 176597, 880125, 4390901, 21920913, 109486993, 547018941, 2733608905, 13662695645, 68294088535, 341399727335, 1706739347095, 8532741458075, 42660172763995, 213287735579135, 1066389745361635, 5331765761680895

Offset: 0

Wolfdieter Lang, Dec 11 1999

Hankel transform is A082761. - Paul Barry, Apr 14 2010

			G.f. = 1 + 3*x + 13*x^2 + 61*x^3 + 295*x^4 + 1447*x^5 + 7151*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
T.-X. He, L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, p 35.

Cf. A000108, A046521, A046714, A082761.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( Sqrt(1-4*x)/(1-5*x) )); // G. C. Greubel, Jul 28 2024

a[ n_] := SeriesCoefficient[ Sqrt[ 1 - 4 x] / (1 - 5 x), {x, 0, n}]; (* Michael Somos, May 25 2014 *)
a[ n_] := Binomial[ 2 n, n] Hypergeometric2F1[ -n, 1, 1/2, -1/4]; (* Michael Somos, May 25 2014 *)

{a(n) = if( n<0, 0, polcoeff( sqrt( 1 - 4*x + x * O(x^n)) / (1 - 5*x), n))}; /* Michael Somos, May 25 2014 */

def A046748_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt(1-4*x)/(1-5*x) ).list()
A046748_list(40) # G. C. Greubel, Jul 28 2024

a(n) = binomial(2*n, n)*Sum_{k=0..n} binomial(n, k)/binomial(2*k, k).
a(n) = 5^n - 2*A046714(n-1), A046714(-1) := 0.
a(n) = 5*a(n-1) - 2*A000108(n-1).
G.f.: sqrt(1-4*x)/(1-5*x).
a(n) = (3*(3*n-2)/n)*a(n-1) - (10*(2*n-3)/n)*a(n-2), n >= 1, a(-1) := 0, a(0)=1 (homogeneous recursion).
a(n) = binomial(2*n,n)*hypergeom([ -n,1 ],[ 1/2 ],-1/4) (hypergeometric 2F1 form).
0 = a(n)*(+400*a(n+1) - 330*a(n+2) + 50*a(n+3)) + a(n+1)*(-30*a(n+1) + 71*a(n+2) - 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, May 25 2014
a(n) ~ 5^(n - 1/2). - Vaclav Kotesovec, Jul 07 2016
D-finite with recurrence n*a(n) +3*(-3*n+2)*a(n-1) +10*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jul 23 2017

A046814 Row sums of triangle A046527.

Original entry on oeis.org

1, 2, 8, 37, 179, 881, 4369, 21746, 108444, 541362, 2704158, 13512392, 67534828, 337584992, 1687627800, 8437136085, 42182258715, 210899507685, 1054456597965, 5272139698215, 26360193558735, 131799177579015

Offset: 0

Wolfdieter Lang

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

Cf. A000108, A046527, A046714, A046748.

[n le 1 select 1 else 5*Self(n-1) - 3*Catalan(n-1)/(2*n-3): n in [1..40]]; // G. C. Greubel, Jul 28 2024

CoefficientList[Series[(1-4*x)*(1-Sqrt[1-4*x])/(2*x*(1-5*x)), {x,0,40}], x] (* G. C. Greubel, Jul 28 2024 *)

@CachedFunction
def A046814(n): return 1 if n==0 else 5*A046814(n-1) - 3*catalan_number(n)/(2*n-1)
[A046814(n) for n in range(41)] # G. C. Greubel, Jul 28 2024

G.f.: c(x) * (1-4*x) / (1-5*x), where c(x) = g.f. for Catalan A000108.
a(n) = C(n) + A046714(n-1) with A046714(-1) = 0 and C(n) = A000108(n) are the Catalan numbers.
a(n) = C(n) + (5^n - A046748(n))/2.
a(n) = 5*a(n-1) - 3*C(n)/(2*n-1), a(0)=1.
D-finite with recurrence a(n) = (9*n-1)*a(n-1)/(n+1) - 10*(2*n-3)*a(n-2)/(n+1), n >= 2, a(0)=1, a(1)=2.

Offset corrected by Sean A. Irvine, Apr 25 2021

A046885 Row sums of triangle A046658.

Original entry on oeis.org

1, 4, 18, 85, 411, 2013, 9933, 49236, 244750, 1218888, 6077644, 30329434, 151439158, 756452890, 3779590010, 18888255205, 94405918355, 471899946985, 2359022096225, 11793343217935, 58960151969255, 294776293579255

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000108, A046658, A046714, A046748.

[n le 1 select 1 else 5*Self(n-1) - Catalan(n-1): n in [1..40]]; // G. C. Greubel, Jul 28 2024

Rest@CoefficientList[Series[Sqrt[1-4*x]*(1-Sqrt[1-4*x])/(2*(1-5*x)), {x,0,40}], x] (* G. C. Greubel, Jul 28 2024 *)

@CachedFunction
def A046885(n): return 1 if n==1 else 5*A046885(n-1) - catalan_number(n-1)
[A046885(n) for n in range(1,41)] # G. C. Greubel, Jul 28 2024

a(n) = 2*5^(n-1) - A046714(n-1) = (A046748(n) - 5^(n-1))/2.
G.f.: x*(2 - c(x))/(1-5*x), where c(x) is the g.f. of A000108 (Catalan numbers).
Inhomogeneous recursion: a(n) = 5*a(n-1) - C(n-1), n >= 2, a(1)=1; C(n) = A000108(n) (Catalan).
Homogeneous recursion: a(n) = (3*(3*n-2)/n)*a(n-1) - (10*(2*n-3)/n)*a(n-2), n >= 3, a(1)=1, a(2)=4.

A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2k)!(x - 1)^(n-k)/((k + 1)!*k!).

Original entry on oeis.org

1, 0, 1, 2, -1, 1, 3, 3, -2, 1, 11, 0, 5, -3, 1, 31, 11, -5, 8, -4, 1, 101, 20, 16, -13, 12, -5, 1, 328, 81, 4, 29, -25, 17, -6, 1, 1102, 247, 77, -25, 54, -42, 23, -7, 1, 3760, 855, 170, 102, -79, 96, -65, 30, -8, 1, 13036, 2905, 685, 68, 181, -175, 161, -95, 38, -9, 1, 45750, 10131, 2220, 617, -113, 356, -336, 256, -133, 47, -10, 1

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

The polynomials C_n(x) have generating function G(x,t) = (1 - sqrt(1 - 4*t))/(2*t*(1 + t - x*t)) = 1 + x*t + (x^2 - x + 2)*t^2 + (x^3 - 2*x^2 + 3*x + 3)*t^3 + ...
C_n(x) can be defined by the recurrence relation C_n(x) = (x - 1)*C_(n-1)(x) + (2n)!/((n + 1)!*n!), C_0(x) = 1 or the equivalent form C_n(x) = (x - 1)*C_(n-1)(x) + C_n(1), C_0(x) = 1.
C_n(x) can be defined as convolution of Catalan numbers and powers of (x - 1).
Discriminants of C_n(x) gives the sequence: 1, 1, -7, -543, 533489, 7080307052, -1318026434480736, -3526797951451513832247, 137992774365121594001729513153, ...
C_n(0) = A032357(n).
C_n(1) = C_n(x) - (x - 1)*C_(n-1)(x) = A000108(n).
C_n(2) = Sum_{m=0..n} C_1(m) = A014137(n).
C_n(3) = A014318(n).
C_n(5) = A000346(n).
C_n(6) = A046714(n).

			Triangle begins:
   1;
   0,  1;
   2, -1,  1;
   3,  3, -2,  1;
  11,  0,  5, -3,  1;
  31, 11, -5,  8, -4,  1;
  ...
The first few polynomials are:
  C_0(x) = 1;
  C_1(x) = x;
  C_2(x) = x^2 -   x   + 2;
  C_3(x) = x^3 - 2*x^2 + 3*x   + 3;
  C_4(x) = x^4 - 3*x^3 + 5*x^2         + 11;
  C_5(x) = x^5 - 4*x^4 + 8*x^3 - 5*x^2 + 11*x + 31;
  ...

G. C. Greubel, Rows n=0..100 of triangle, flattened
Ilya Gutkovskiy, Polynomials C_n(x)
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A130595.

CoefficientList[RecurrenceTable[{c[0] == 1, c[n] == (x - 1) c[n - 1] + CatalanNumber[n]}, c, {n, 11}], x]
T[n_, n_]:= 1; T[n_, 0]:= (-1)^n*Sum[CatalanNumber[k]*(-1)^k, {k, 0, n}]; T[n_, k_]:= T[n - 1, k - 1] - T[n - 1, k]; Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 04 2018 *)

{T(n, k) = if(k==n, 1, if(k==0, sum(j=0,n, (-1)^(n-j)*(2*j)!/(j!*(j+1)!)), T(n-1, k-1) - T(n-1, k))) };
for(n=0, 10, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 04 2018

For triangle: T(n,n)=1, T(n,0) = Sum_{k=0..n} (-1)^(n-k)*(2*k)!/(k! * (k+1)!), T(n, k) = T(n-1, k-1) - T(n-1, k). - G. C. Greubel, Nov 04 2018

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A046748 Row sums of triangle A046521.

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A046814 Row sums of triangle A046527.

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A046885 Row sums of triangle A046658.

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A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2*k)!*(x - 1)^(n-k)/((k + 1)!*k!).

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A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2k)!(x - 1)^(n-k)/((k + 1)!*k!).