A063886 -id:A063886 - cp's OEIS frontend

A130777 Coefficients of first difference of Chebyshev S polynomials.

1, -1, 1, -1, -1, 1, 1, -2, -1, 1, 1, 2, -3, -1, 1, -1, 3, 3, -4, -1, 1, -1, -3, 6, 4, -5, -1, 1, 1, -4, -6, 10, 5, -6, -1, 1, 1, 4, -10, -10, 15, 6, -7, -1, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, -1, -5, 15, 20, -35, -21, 28, 8, -9, -1, 1, 1, -6, -15, 35, 35, -56, -28, 36, 9, -10, -1, 1

Offset: 0

Philippe Deléham, Jul 14 2007

Inverse of triangle in A061554.
Signed version of A046854.
From Paul Barry, May 21 2009: (Start)
Riordan array ((1-x)/(1+x^2),x/(1+x^2)).
This triangle is the coefficient triangle for the Hankel transforms of the family of generalized Catalan numbers that satisfy a(n;r)=r*a(n-1;r)+sum{k=1..n-2, a(k)*a(n-1-k;r)}, a(0;r)=a(1;r)=1. The Hankel transform of a(n;r) is h(n)=sum{k=0..n, T(n,k)*r^k} with g.f. (1-x)/(1-r*x+x^2). These sequences include A086246, A000108, A002212. (End)
From Wolfdieter Lang, Jun 11 2011: (Start)
The Riordan array ((1+x)/(1+x^2),x/(1+x^2)) with entries Phat(n,k)= ((-1)^(n-k))*T(n,k) and o.g.f. Phat(x,z)=(1+z)/(1-x*z+z^2) for the row polynomials Phat(n,x) is related to Chebyshev C and S polynomials as follows.
Phat(n,x) = (R(n+1,x)-R(n,x))/(x+2) = S(2*n,sqrt(2+x))
  with R(n,x)=C_n(x) in the Abramowitz and Stegun notation, p. 778, 22.5.11. See A049310 for the S polynomials. Proof from the o.g.f.s.
Recurrence for the row polynomials Phat(n,x):
  Phat(n,x) = x*Phat(n-1,x) - Phat(n-2,x) for n>=1; Phat(-1,x)=-1, Phat(0,x)=1.
The A-sequence for this Riordan array Phat (see the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices) is given by 1, 0, -1, 0, -1, 0, -2, 0, -5,.., starting with 1 and interlacing the negated A000108 with zeros (o.g.f. 1/c(x^2) = 1-c(x^2)*x^2, with the o.g.f. c(x) of A000108).
The Z-sequence has o.g.f. sqrt((1-2*x)/(1+2*x)), and it is given by A063886(n)*(-1)^n.
The A-sequence of the Riordan array T(n,k) is identical with the one for the Riordan array Phat, and the Z-sequence is -A063886(n).
(End)
The row polynomials P(n,x) are the characteristic polynomials of the adjacency matrices of the graphs which look like P_n (n vertices (nodes), n-1 lines (edges)), but vertex no. 1 has a loop. - Wolfdieter Lang, Nov 17 2011
From Wolfdieter Lang, Dec 14 2013: (Start)
The zeros of P(n,x) are x(n,j) = -2*cos(2*Pi*j/(2*n+1)), j=1..n. From P(n,x) = (-1)^n*S(2*n,sqrt(2-x)) (see, e.g., the Lemma 6 of the W. Lang link).
The discriminants of the P-polynomials are given in A052750. (End)

			The triangle T(n,k) begins:
n\k  0   1   1   3    4    5    6    7    8    9  10  11  12  13 14 15 ...
0:   1
1:  -1   1
2:  -1  -1   1
3:   1  -2  -1   1
4:   1   2  -3  -1    1
5:  -1   3   3  -4   -1    1
6:  -1  -3   6   4   -5   -1    1
7:   1  -4  -6  10    5   -6   -1    1
8:   1   4 -10 -10   15    6   -7   -1    1
9:  -1   5  10 -20  -15   21    7   -8   -1    1
10: -1  -5  15  20  -35  -21   28    8   -9   -1   1
11:  1  -6 -15  35   35  -56  -28   36    9  -10  -1   1
12:  1   6 -21 -35   70   56  -84  -36   45   10 -11  -1   1
13: -1   7  21 -56  -70  126   84 -120  -45   55  11 -12  -1   1
14: -1  -7  28  56 -126 -126  210  120 -165  -55  66  12 -13  -1  1
15:  1  -8 -28  84  126 -252 -210  330  165 -220 -66  78  13 -14 -1  1
...  reformatted and extended - _Wolfdieter Lang_, Jul 31 2014.
---------------------------------------------------------------------------
From _Paul Barry_, May 21 2009: (Start)
Production matrix is
-1, 1,
-2, 0, 1,
-2, -1, 0, 1,
-4, 0, -1, 0, 1,
-6, -1, 0, -1, 0, 1,
-12, 0, -1, 0, -1, 0, 1,
-20, -2, 0, -1, 0, -1, 0, 1,
-40, 0, -2, 0, -1, 0, -1, 0, 1,
-70, -5, 0, -2, 0, -1, 0, -1, 0, 1 (End)
Row polynomials as first difference of S polynomials:
P(3,x) = S(3,x) - S(2,x) = (x^3 - 2*x) - (x^2 -1) = 1 - 2*x - x^2 +x^3.
Alternative triangle recurrence (see a comment above): T(6,2) = T(5,2) + T(5,1) = 3 + 3 = 6. T(6,3) = -T(5,3) + 0*T(5,1) = -(-4) = 4. - _Wolfdieter Lang_, Jul 31 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available).

Hyeong-Kwan Ju, On the sequence generated by a certain type of matrices, Honam Math. J. 39, No. 4, 665-675 (2017), Theorem 2.16.
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012-2017; see Definition 1, Lemma 6 and Remark 4.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A066170, A046854, A057077 (first column).

Row sums: A010892(n+1); repeat(1,0,-1,-1,0,1). Alternating row sums: A061347(n+2); repeat(1,-2,1).

A130777 := proc(n,k): (-1)^binomial(n-k+1,2)*binomial(floor((n+k)/2),k) end: seq(seq(A130777(n,k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

T[n_, k_] := (-1)^Binomial[n - k + 1, 2]*Binomial[Floor[(n + k)/2], k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2017, from Maple *)

@CachedFunction
def A130777(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = A130777(n-1,k) if n==1 else 0
    return A130777(n-1,k-1) - A130777(n-2,k) - h
for n in (0..9): [A130777(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Number triangle T(n,k) = (-1)^C(n-k+1,2)*C(floor((n+k)/2),k). - Paul Barry, May 21 2009
From Wolfdieter Lang, Jun 11 2011: (Start)
Row polynomials: P(n,x) = sum(k=0..n, T(n,k)*x^k) = R(2*n+1,sqrt(2+x)) / sqrt(2+x), with Chebyshev polynomials R with coefficients given in A127672 (scaled T-polynomials).
R(n,x) is called C_n(x) in Abramowitz and Stegun's handbook, p. 778, 22.5.11.
P(n,x) = S(n,x)-S(n-1,x), n>=0, S(-1,x)=0, with the Chebyshev S-polynomials (see the coefficient triangle A049310).
O.g.f. for row polynomials: P(x,z):= sum(n>=0, P(n,x)*z^n ) = (1-z)/(1-x*z+z^2).
  (from the o.g.f. for R(2*n+1,x), n>=0, computed from the o.g.f. for the R-polynomials (2-x*z)/(1-x*z+z^2) (see A127672))
Proof of the Chebyshev connection from the o.g.f. for Riordan array property of this triangle (see the P. Barry comment above).
For the A- and Z-sequences of this Riordan array see a comment above. (End)
abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) T(n,n-k) = A108299(n,k); abs(T(n,n-k)) = A065941(n,k). - Johannes W. Meijer, Aug 08 2011
From Wolfdieter Lang, Jul 31 2014: (Start)
Similar to the triangles A157751, A244419 and A180070 one can give for the row polynomials P(n,x) besides the usual three term recurrence another one needing only one recurrence step. This uses also a negative argument, namely P(n,x) = (-1)^(n-1)*(-1 + x/2)*P(n-1,-x) + (x/2)*P(n-1,x), n >= 1, P(0,x) = 1. Proof by computing the o.g.f. and comparing with the known one. This entails the alternative triangle recurrence T(n,k) = (-1)^(n-k)*T(n-1,k) + (1/2)*(1 + (-1)^(n-k))*T(n-1,k-1), n >= m >= 1, T(n,k) = 0 if n < k and T(n,0) = (-1)^floor((n+1)/2) = A057077(n+1). [P(n,x) recurrence corrected Aug 03 2014]
(End)

New name and Chebyshev comments by Wolfdieter Lang, Jun 11 2010

A084099 Expansion of (1+x)^2/(1+x^2).

Original entry on oeis.org

1, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0

Offset: 0

Paul Barry, May 15 2003

Inverse binomial transform of A077860. Partial sums of A084100.
Transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x)->((1-x^2)/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 12 2004
Euler transform of length 4 sequence [2, -3, 0, 1]. - Michael Somos, Aug 04 2009

			G.f. = 1 + 2*x - 2*x^3 + 2*x^5 - 2*x^7 + 2*x^9 - 2*x^11 + 2*x^13 - 2*x^15 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28.
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A063886, A077860, A084100, A101455, A180969.

[1] cat [Integers()!((1-(-1)^n)*(-1)^(n*(n-1)/2)): n in [1..100]]; // Wesley Ivan Hurt, Oct 27 2015

A084099:=n->(1-(-1)^n)*(-1)^((2*n-1+(-1)^n)/4): 1,seq(A084099(n), n=1..100); # Wesley Ivan Hurt, Oct 27 2015

CoefficientList[Series[(1+x)^2/(1+x^2),{x,0,110}],x] (* or *) Join[ {1}, PadRight[{},120,{2,0,-2,0}]] (* Harvey P. Dale, Nov 23 2011 *)

{a(n) = if( n<1, n==0, 2 * if( n%2, (-1)^(n\2)) )}; /* Michael Somos, Aug 04 2009 */

a(n) = if(n==0, 1, I*((-I)^n-I^n)) \\ Colin Barker, Oct 27 2015

Vec((1+x)^2/(1+x^2) + O(x^100)) \\ Colin Barker, Oct 27 2015

G.f.: (1+x)^2/(1+x^2).
a(n) = 2 * A101455(n) for n>0. - N. J. A. Sloane, Jun 01 2010
a(n+2) = (-1)^A180969(1,n)*((-1)^n - 1). - Adriano Caroli, Nov 18 2010
G.f.: 4*x + 2/(1+x)/G(0), where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 19 2013
From Wesley Ivan Hurt, Oct 27 2015: (Start)
a(n) = (1-sign(n)*(-1)^n)*(-1)^floor(n/2).
a(n) = 2*(n mod 2)*(-1)^floor(n/2) for n>0, a(0)=1.
a(n) = (1-(-1)^n)*(-1)^(n*(n-1)/2) for n>0, a(0)=1. (End)
From Colin Barker, Oct 27 2015: (Start)
a(n) = -a(n-2).
a(n) = i*((-i)^n-i^n) for n>0, where i = sqrt(-1).
(End)

A300788 Number of strict integer partitions of n in which the even parts appear as often at even positions as at odd positions.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 19, 23, 26, 30, 35, 42, 47, 54, 62, 73, 82, 94, 107, 124, 139, 158, 179, 206, 230, 260, 293, 334, 372, 420, 470, 532, 591, 664, 740, 835, 924, 1034, 1148, 1288, 1422, 1588, 1756, 1962, 2161, 2404

Offset: 0

Gus Wiseman, Mar 12 2018

nonn

			The a(9) = 3 strict partitions: (9), (621), (531). Missing are: (81), (72), (63), (54), (432).

Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 291 terms from Fausto A. C. Cariboni)

Cf. A000712, A000898, A001405, A026010, A045931, A063886, A097613, A130780, A171966, A239241, A300787, A300789.

cobal[y_]:=Sum[(-1)^x,{x,Join@@Position[y,_?EvenQ]}];
Table[Length[Select[IntegerPartitions[n],cobal[#]===0&&UnsameQ@@#&]],{n,0,40}]

a(41)-a(58) from Alois P. Heinz, Mar 13 2018

A360318 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 12, 74, 466, 2982, 19320, 126390, 833220, 5527190, 36852052, 246751854, 1658106394, 11176100138, 75528743352, 511600414554, 3472363279170, 23609924743590, 160788499672020, 1096566516149790, 7488135911236806, 51193972101241362, 350368409215623192

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360317, A360319, A360321, A360322.

Cf. A085364, A122898.

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

my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)))

G.f.: sqrt( (1-3*x)/(1-7*x) ).
n*a(n) = 2*(5*n-4)*a(n-1) - 21*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/3)^i * a(j) * a(i-j) = (7/3)^n.
a(n) = 2 * A122898(n-1) for n > 0.
a(n) ~ 2 * 7^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A028330 Elements to the right of the central elements of the even-Pascal triangle A028326.

Original entry on oeis.org

2, 2, 6, 2, 8, 2, 20, 10, 2, 30, 12, 2, 70, 42, 14, 2, 112, 56, 16, 2, 252, 168, 72, 18, 2, 420, 240, 90, 20, 2, 924, 660, 330, 110, 22, 2, 1584, 990, 440, 132, 24, 2, 3432, 2574, 1430, 572, 156, 26, 2, 6006, 4004, 2002, 728, 182, 28, 2, 12870, 10010, 6006, 2730

Offset: 0

Mohammad K. Azarian

			This sequence represents the following portion of A028330(n,k), with x being the elements of A028329(n):
  x;
  .,  2;
  .,  x,  2;
  .,  .,  6,  2;
  .,  .,  x,  8,  2;
  .,  .,  ., 20, 10,   2;
  .,  .,  .,  x, 30,  12,   2;
  .,  .,  .,  ., 70,  42,  14,    2;
  .,  .,  .,  .,  x, 112,  56,   16,   2;
  .,  .,  .,  .,  ., 252, 168,   72,  18,   2;
  .,  .,  .,  .,  .,   x, 420,  240,  90,  20,   2;
  .,  .,  .,  .,  .,   ., 924,  660, 330, 110,  22,  2;
  .,  .,  .,  .,  .,   .,   x, 1584, 990, 440, 132, 24, 2;
As an irregular triangle:
    2;
    2;
    6,   2;
    8,   2;
   20,  10,   2;
   30,  12,   2;
   70,  42,  14,   2;
  112,  56,  16,   2;
  252, 168,  72,  18,  2;
  420, 240,  90,  20,  2;
  924, 660, 330, 110, 22,  2;

G. C. Greubel, Table of n, a(n) for n = 0..2549

Cf. A028326, A028327, A028328, A028329, A028331, A028332.

Cf. A001405, A014413, A058622, A063886, A202736.

[[2*Binomial(n,k): k in [Floor((n+2)/2)..n]]: n in [1..12]]; // G. C. Greubel, Jul 14 2024

Table[2*Binomial[n+1, k+1 +Floor[(n+1)/2]], {n,0,12}, {k,0,Floor[n/2] }]//Flatten (* G. C. Greubel, Jul 14 2024 *)

def A028326(n,k): return 2*binomial(n, k)
flatten([[A028326(n,k) for k in range(((n+2)//2), n+1)] for n in range(1,21)]) # G. C. Greubel, Jul 14 2024

a(n) = 2 * A014413(n). - Sean A. Irvine, Dec 29 2019
From G. C. Greubel, Jul 14 2024: (Start)
T(n, k) = 2*binomial(n+1, k+1 + floor((n+1)/2)) for n >= 0, 0 <= k <= floor(n/2).
Sum_{k=0..floor(n/2)} T(n, k) = A202736(n+1) = 2*A058622(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n, k) = 2*A001405(n) = A063886(n+1). (End)

More terms from James Sellers

A047073 a(n) = Sum_{j=0..n} A047072(j, n-j).

Original entry on oeis.org

1, 2, 4, 4, 8, 12, 24, 40, 80, 140, 280, 504, 1008, 1848, 3696, 6864, 13728, 25740, 51480, 97240, 194480, 369512, 739024, 1410864, 2821728, 5408312, 10816624, 20801200, 41602400, 80233200, 160466400, 310235040, 620470080, 1202160780

Offset: 0

Clark Kimberling

nonn

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

Cf. A047072, A063886.

[n le 1 select n+1 else 4*Binomial(n-2, Floor((n-2)/2)): n in [0..40]]; // G. C. Greubel, Oct 13 2022

Table[If[n<2, n+1, 4*Binomial[n-2, Floor[(n-2)/2]]], {n,0,40}] (* G. C. Greubel, Oct 13 2022 *)

a(n) = if(n<2, max(0,n+1), 4*binomial(n-2, n\2-1))

[4*binomial(n-2, ((n-2)//2)) + (n+1)*int(n<2) for n in range(41)] # G. C. Greubel, Oct 13 2022

a(n) = 2*A063886(n-1) + (n+1)*[n<2].
G.f.: 1 + 2*x*sqrt((1+2*x)/(1-2*x)). - Michael Somos
E.g.f.: 1 - x*BesselI(1, 2*x)*(2 + Pi*(1 + 2*x)*StruveL(0, 2*x)) + x*(1 + 2*x)*BesselI(0, 2*x)*(2 + Pi*StruveL(1, 2*x)). - Stefano Spezia, May 11 2024

A265870 T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with no element plus any antidiagonal neighbor equal to n-1.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 4, 42, 24, 1, 6, 340, 984, 120, 1, 12, 3136, 56496, 48216, 720, 1, 20, 30456, 3965392, 28645840, 2869200, 5040, 1, 40, 307464, 307543008, 21193260208, 20091622560, 278125200, 40320, 1, 70, 3182784, 25346247472, 17674408924672

Offset: 1

R. H. Hardin, Dec 17 2015

Table starts
......1.............1.................1...................1...................1
......2.............2.................4...................6..................12
......6............42...............340................3136...............30456
.....24...........984.............56496.............3965392...........307543008
....120.........48216..........28645840.........21193260208......17674408924672
....720.......2869200.......20091622560.....187225073253120.2036853990307247808
...5040.....278125200....29056095714720.4173639524545757568
..40320...31015797120.52981034285587200
.362880.4991834736000

			Some solutions for n=3 k=4
..0..2..0..0....0..0..1..2....1..2..0..0....1..2..2..1....0..2..0..0
..1..1..1..1....1..2..1..0....2..1..1..2....1..1..2..0....1..1..1..1
..2..2..2..0....1..2..0..2....0..2..1..0....0..2..0..0....2..2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..60

Column 1 is A000142.

Row 2 is A063886.

A303537 Expansion of ((1 + 4x)/(1 - 4x))^(1/4).

Original entry on oeis.org

1, 2, 2, 12, 22, 124, 276, 1496, 3686, 19436, 51068, 263720, 724860, 3681880, 10466920, 52450992, 153093254, 758495564, 2261603564, 11096526344, 33676743956, 163842737928, 504738342808, 2437418983888, 7605947276508, 36487283224952, 115140704639576

Offset: 0

Seiichi Manyama, Apr 25 2018

nonn

Let ((1 + k*x)/(1 - k*x))^(1/k) = a(0) + a(1)*x + a(2)*x^2 + ...

Then n*a(n) = 2*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A063886, A081654, A151403, A303538.

CoefficientList[Series[Surd[(1+4x)/(1-4x),4],{x,0,40}],x] (* Harvey P. Dale, Jul 25 2021 *)

N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(1/4))

a(n) ~ 2^(2*n + 1/4) / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 26 2018
n*a(n) = 2*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.
G.f.: A(x)=F(x*G(x^2)), where F(x) is the g.f. for A063886, and G(x) is the g.f. for A151403. - Alexander Burstein, Nov 13 2023

A360317 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 10, 52, 278, 1516, 8388, 46920, 264678, 1503052, 8581676, 49215256, 283297660, 1635904376, 9472214344, 54975423504, 319729353606, 1862896455180, 10871759717916, 63539265366264, 371837338366740, 2178604586281128, 12778264475444280, 75022726995053808

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360318, A360319, A360321, A360322.

Cf. A005573, A085363.

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

my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)))

G.f.: sqrt( (1-2*x)/(1-6*x) ).
n*a(n) = 2*(4*n-3)*a(n-1) - 12*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/2)^i * a(j) * a(i-j) = 3^n.
a(n) = 2 * A005573(n-1) for n > 0.
a(n) ~ 2^(n + 1/2) * 3^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/2)^n * Sum_{k=0..n} 3^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A106180 Matrix inverse of number triangle A046854.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 0, 2, -2, -1, 1, -2, 2, 3, -3, -1, 1, 0, -5, 5, 4, -4, -1, 1, 5, -5, -9, 9, 5, -5, -1, 1, 0, 14, -14, -14, 14, 6, -6, -1, 1, -14, 14, 28, -28, -20, 20, 7, -7, -1, 1, 0, -42, 42, 48, -48, -27, 27

Offset: 0

Paul Barry, Apr 24 2005

First column is A105523; second column is A106181.
Triangle T(n,k), 0 <= k <= n, read by rows given by [ -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 29 2006
A124448*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 16 2007

			Triangle begins
   1;
  -1,  1;
   0, -1,  1;
   1, -1, -1,  1;
   0,  2, -2, -1,  1;
  -2,  2,  3, -3, -1,  1;
   0, -5,  5,  4, -4, -1,  1;

Cf. A000108.

Riordan array (1-y, y) where y=-(1-sqrt(1+4x^2))/(2x).
Sum_{k=0..n} abs(T(n,k)) = A063886(n). - Philippe Deléham, Oct 06 2006
T(0,0)=1; T(n,k)=0 if k < 0 or if k > n; T(n,0) = -T(n-1,0) - T(n-1,1); T(n,k) = T(n,k-1) - T(n-1,k+1) for k >= 1. - Philippe Deléham, Oct 27 2007
T(2n,0) = A000007(n); T(2n+2,2k+2) = -T(2n+2,2k+1) = (-1)^(n-k)*A039598(n,k); T(2n+1,2k+1) = -T(2n+1,2k) = (-1)^(n-k)*A039599(n,k). - Philippe Deléham, Oct 29 2007
Sum_{k>=0} T(m,k)*T(n,k)*(-1)^k = T(m+n,0) = A105523(m+n). - Philippe Deléham, Jan 24 2010

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