A123876 -id:A123876 - cp's OEIS frontend

A116395 Riordan array (1/sqrt(1-4x), (1/sqrt(1-4x)-1)/2).

1, 2, 1, 6, 5, 1, 20, 22, 8, 1, 70, 93, 47, 11, 1, 252, 386, 244, 81, 14, 1, 924, 1586, 1186, 500, 124, 17, 1, 3432, 6476, 5536, 2794, 888, 176, 20, 1, 12870, 26333, 25147, 14649, 5615, 1435, 237, 23, 1, 48620, 106762, 112028, 73489, 32714, 10135, 2168, 307, 26, 1

Offset: 0

Paul Barry, Feb 12 2006

Row sums are A007854. Diagonal sums are A116396.
Triangle T(n,k), 0 <= k <= n, read by rows given by [2,1,1,1,1,1,1,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 05 2007
Inverse of Riordan array (1/(1+2*x), x*(1+x)/(1+2*x)^2) (see A123876). - Philippe Deléham, Oct 25 2007

			Triangle begins:
    1;
    2,   1;
    6,   5,   1;
   20,  22,   8,  1;
   70,  93,  47, 11,  1;
  252, 386, 244, 81, 14, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Joseph Pappe, Digjoy Paul and Anne Schilling, An area-depth symmetric q,t-Catalan polynomial, arXiv:2109.06300 [math.CO], 2021. See Remark 2.4 p. 4.

[[ Round((4^n/2^k)*(&+[ (-1)^(k-j)*Binomial(k, j)*Gamma(n+(j+1)/2)/(Factorial(n)*Gamma((j+1)/2)) : j in [0..k]])) : k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 28 2019

T[n_, k_]:= (4^n/2^k)*Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n+(j-1)/2, n], {j, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 28 2019 *)

{T(n,k) = (4^n/2^k)*sum(j=0, k, (-1)^(k-j)*binomial(k, j)* binomial(n+(j-1)/2, n))}; \\ G. C. Greubel, May 28 2019

[[(4^n/2^k)*sum( (-1)^(k-j)*binomial(k, j)* binomial(n+(j-1)/2, n) for j in (0..k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 28 2019

Number triangle T(n,k) = (4^n/2^k)*Sum_{j=0..k} C(k,j)*C(n+(j-1)/2,n)*(-1)^(k-j).
Sum_{k=0..n} (-1)^k*T(n,k) = A000108(n), Catalan numbers. - Philippe Deléham, Nov 07 2006
T(n,k) = Sum_{j>=0} A039599(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007
Sum_{k=0..n} T(n,k)*x^k = A127053(n), A126985(n), A127016(n), A127017(n), A126987(n), A126986(n), A126982(n), A126984(n), A126983(n), A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for x = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Oct 25 2007

A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 18, 8, 1, 16, 56, 41, 11, 1, 32, 160, 170, 73, 14, 1, 64, 432, 620, 377, 114, 17, 1, 128, 1120, 2072, 1666, 704, 164, 20, 1, 256, 2816, 6496, 6608, 3649, 1178, 223, 23, 1, 512, 6912, 19392, 24192, 16722, 7001, 1826, 291, 26, 1, 1024, 16640, 55680, 83232, 69876, 36365, 12235, 2675, 368, 29, 1

Offset: 0

Paul Barry, Nov 15 2005

Row sums are A081567. Diagonal sums are A085810. Product of Pascal triangle A007318 and Morgan-Voyce triangle A085478.

Unsigned version of A123876. - Philippe Deléham, Oct 25 2007

			Triangle begins:
   1;
   2,   1;
   4,   5,   1;
   8,  18,   8,  1;
  16,  56,  41, 11,  1;
  32, 160, 170, 73, 14, 1;
  ...

Cf. A081567, A085810, A007318, A085478, A123876.

T(2n,n) gives A026000.

Number triangle T(n,k) = Sum_{j=0..n} C(n, j)*C(j+k, 2k);
T(n,k) = Sum_{j=0..n} C(n, k+j)*C(k, k-j)*2^(n-k-j);
T(n,k) = Sum_{j=0..n-k} C(n+k-j, n-k-j)*C(k, j)*(-1)^j*2^(n-k-j).
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 4*T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014

More terms from Michel Marcus, Sep 09 2024

A123877 Expansion of (1+2x)/(1+3x+3*x^2).

Original entry on oeis.org

1, -1, 0, 3, -9, 18, -27, 27, 0, -81, 243, -486, 729, -729, 0, 2187, -6561, 13122, -19683, 19683, 0, -59049, 177147, -354294, 531441, -531441, 0, 1594323, -4782969, 9565938, -14348907, 14348907, 0, -43046721

Offset: 0

Paul Barry, Oct 16 2006

Row sums of number triangle A123876.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-3).

Cf. A123876.

a:=[1,-1];; for n in [3..40] do a[n]:=-3*(a[n-1]+a[n-2]); od; a; # G. C. Greubel, Aug 08 2019

I:=[1,-1]; [n le 2 select I[n] else -3*(Self(n-1)+Self(n-2)): n in [1..30]]; // G. C. Greubel, Aug 08 2019

seq(coeff(series((1+2*x)/(1+3*x+3*x^2), x, n+1), x, n), n = 0..40); # G. C. Greubel, Aug 08 2019

CoefficientList[Series[(1+2x)/(1+3x+3x^2),{x,0,40}],x] (* or *) LinearRecurrence[{-3,-3},{1,-1},40] (* Harvey P. Dale, Dec 17 2017 *)

my(x='x+O('x^40)); Vec((1+2*x)/(1+3*x+3*x^2)) \\ G. C. Greubel, Aug 08 2019

def A123877_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+2*x)/(1+3*x+3*x^2)).list()
A123877_list(40) # G. C. Greubel, Aug 08 2019

a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(cos(2*Pi*k/3) + sin(2*Pi*k/3)/sqrt(3)).
G.f.: G(0)*(1+2*x)/(2+3*x), where G(k)= 1 + 1/(1 - x*(k+3)/(x*(k+4) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013
E.g.f.: (1/3)*exp(-3*x/2)*(3*cos((sqrt(3)*x)/2) + sqrt(3)*sin((sqrt(3)*x)/2)). - Stefano Spezia, Aug 08 2019

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A116395 Riordan array (1/sqrt(1-4x), (1/sqrt(1-4x)-1)/2).

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A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

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A123877 Expansion of (1+2x)/(1+3x+3*x^2).

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cp's OEIS Frontend

A116395 Riordan array (1/sqrt(1-4*x), (1/sqrt(1-4*x)-1)/2).

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A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

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A123877 Expansion of (1+2*x)/(1+3*x+3*x^2).

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A116395 Riordan array (1/sqrt(1-4x), (1/sqrt(1-4x)-1)/2).

A123877 Expansion of (1+2x)/(1+3x+3*x^2).