A128014 -id:A128014 - cp's OEIS frontend

A126869 a(n) = Sum_{k = 0..n} binomial(n,floor(k/2))*(-1)^(n-k).

1, 0, 2, 0, 6, 0, 20, 0, 70, 0, 252, 0, 924, 0, 3432, 0, 12870, 0, 48620, 0, 184756, 0, 705432, 0, 2704156, 0, 10400600, 0, 40116600, 0, 155117520, 0, 601080390, 0, 2333606220, 0, 9075135300, 0, 35345263800, 0, 137846528820, 0, 538257874440, 0, 2104098963720, 0, 8233430727600, 0, 32247603683100, 0, 126410606437752, 0

Offset: 0

Philippe Deléham, Mar 16 2007

Hankel transform is 2^n. Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.
From Andrew V. Sutherland, Feb 29 2008: (Start)
Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}.
Moment sequence of the trace of a random matrix in G = SO(2). If X = tr(A) is a random variable (A distributed with Haar measure on G), then a(n) = E[X^n].
Also the moment sequence of the trace of the k-th power of a random matrix in USp(2) = SU(2), for all k > 2.
(End)
From Paul Barry, Aug 10 2009: (Start)
The Hankel transform of 0,1,0,2,0,6,... is 0,-1,0,4,0,-16,0,... with general term I*(-4)^(n/2)(1 - (-1)^n)/4, I = sqrt(-1).
The Hankel transform of 1,1,0,2,0,6,... (which has g.f. 1 + x/sqrt(1 - 4*x^2)) is A164111. (End)
a(n) = A204293(2*n,n): central terms of the triangle in A204293. - Reinhard Zumkeller, Jan 14 2012
a(n) is the total number of closed walks (round trips) of length n on the graph P_N (a line with N nodes and N-1 edges), divided by N, in the limit N -> infinity. See a comment on A198632 and a link under A201198. - Wolfdieter Lang, Oct 10 2012

			a(4) = 6 {UUDD,UDUD,UDDU,DUUD,DUDU,DDUU}.

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 99.

This is A000984 with interspersed zeros. m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

Cf. A107430, A061554, A126120.

a126869 n = a204293_row (2*n) !! n  -- Reinhard Zumkeller, Jan 14 2012

seq((-1)^(n/2)*pochhammer(-n,n/2)/(n/2)!, n=0..43); # Peter Luschny, May 17 2013
seq(n!*coeff(series(hypergeom([],[1],x^2),x,n+1),x,n),n=0..42); # Peter Luschny, Jan 31 2015

Table[(-1)^Floor[n/2] HypergeometricPFQ[{-n,-n},{1},-1],{n,0,30}] (* Peter Luschny, Nov 01 2011 *)

A126869 = lambda n: (2^(n-1)*((-1)^n+1)*gamma((n+1)/2))/(sqrt(pi)*gamma((n+2)/2))
[A126869(n) for n in range(44)] # Peter Luschny, Sep 10 2014

From Andrew V. Sutherland, Feb 29 2008: (Start)
a(2*n) = binomial(2*n,n) = A000984(n); a(2*n+1) = 0.
a(n) = Sum_{k = 0..n} A107430(n,k)*(-1)^(n-k).
a(n) = Sum_{k = 0..n} A061554(n,k)*(-1)^k.
a(n) = (1/Pi)*Integral_{t = 0..Pi} cos^n(t) dt. (End)
E.g.f.: I_0(2*x) where I_n(x) is the modified Bessel function as a function of x. - Benjamin Phillabaum, Mar 10 2011
G.f.: A(x) = 1/sqrt(1 - 4*x^2). - Vladimir Kruchinin, Apr 16 2011
a(n) = (1/Pi)*Integral{x = -2..2} x^n/sqrt((2 - x)*(2 + x)). - Peter Luschny, Sep 12 2011
a(n) = (-1)^floor(n/2) * Hypergeometric([-n,-n],[1], -1). - Peter Luschny, Nov 01 2011
E.g.f.: E(0)/(1 - x) where E(k) = 1 - x/(1 - x/(x - (k+1)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
E.g.f.: 1 + x^2/(Q(0) - x^2), where Q(k) = x^2 + (k+1)^2 - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013
G.f.: 1/(1 - 2*x^2*Q(0)), where Q(k) = 1 + (4*k+1)*x^2/(k+1 - x^2*(2*k+2)*(4*k+3)/(2*x^2*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: G(0)/(1+x), where G(k) = 1 + x*(2+5*x)*(4*k+1)/((4*k+2)*(1+x)^2 - 2*(2*k+1)*(4*k+3)*x*(2+5*x)*(1+x)^2/((4*k+3)*x*(2+5*x) + 4*(k+1)*(1+x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 19 2014
a(n) = 2^n*JacobiP(n,0,-1/2-n,-3). - Peter Luschny, Aug 02 2014
a(n) = (2^(n-1)*((-1)^n+1)*Gamma((n+1)/2))/(sqrt(Pi)*Gamma((n+2)/2)). - Peter Luschny, Sep 10 2014
a(n) = n!*[x^n]hypergeom([],[1],x^2). - Peter Luschny, Jan 31 2015
a(n) = 2^n*hypergeom([1/2,-n],[1],2). - Peter Luschny, Feb 03 2015
From Peter Bala, Jul 25 2016: (Start)
a(n) = (-1)^floor(n/2)*Sum_{k = 0..n} (-1)^k*binomial(n,k)^2.
D-finite with recurrence: a(n) = 4*(n - 1)/n * a(n-2) with a(0) = 1, a(1) = 0. (End)
From Ilya Gutkovskiy, Jul 25 2016: (Start)
Inverse binomial transform of A002426.
a(n) = Sum_{k=0..n} (-1)^k*A128014(k).
a(n) ~ 2^n*((-1)^n + 1)/sqrt(2*Pi*n). (End)

A128018 Expansion of (1-4x)/(1-2x+4*x^2).

Original entry on oeis.org

1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728, 268435456, 1073741824, 1073741824, -2147483648, -8589934592

Offset: 0

Paul Barry, Feb 11 2007

Hankel transform of A128014(n+1). Binomial transform of A128019.
Hankel transform of A002426(n+1). - Paul Barry, Mar 15 2008
Hankel transform of A007971(n+1). - Paul Barry, Sep 30 2009
Hankel transform of A103970 is a(n)/4^C(n+1,2). - Paul Barry, Nov 20 2009
The real part of Q^(n+1), where Q is the quaternion 1+i+j+k. - Stanislav Sykora, Jun 11 2012.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-4).

Cf. A002426, A007971, A128014, A103970, A128019, A138340.

CoefficientList[Series[(1 - 4*x)/(1 - 2*x + 4*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{2,-4},{1,-2},50] (* G. C. Greubel, Feb 28 2017 *)

x='x+O('x^50); Vec((1-4*x)/(1-2*x+4*x^2)) \\ G. C. Greubel, Feb 28 2017

a(n) = A138340(n)/2^n. - Philippe Deléham, Nov 14 2008
a(n) = 2^(n+1)*cos(Pi*(n+1)/3). - Richard Choulet, Nov 19 2008
From Paul Barry, Oct 21 2009: (Start)
a(n) = Sum_{k=0..floor((n+1)/2)} C(n+1,2*k)*(-3)^k.
a(n) = ((1+i*sqrt(3))^(n+1) + (1-i*sqrt(3))^(n+1))/2, i=sqrt(-1). (End)
G.f.: G(0)/(2*x)-1/x, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = 2^n*A057079(n+2). - R. J. Mathar, Mar 04 2018
Sum_{n>=0} 1/a(n) = 1/3. - Amiram Eldar, Feb 14 2023

A249095 Triangle read by rows: interleaving successive pairs of rows of Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 4, 3, 6, 3, 4, 1, 1, 1, 1, 5, 4, 10, 6, 10, 4, 5, 1, 1, 1, 1, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1, 1, 1, 1, 7, 6, 21, 15, 35, 20, 35, 15, 21, 6, 7, 1, 1, 1, 1, 8, 7, 28, 21, 56, 35, 70, 35, 56, 21, 28, 7, 8, 1, 1

Offset: 0

Reinhard Zumkeller, Nov 14 2014

Length of row n = 2*n+1;
T(n,2*k) = A007318(n,k), 0 <= k <= n;
T(n,2*k+1) = A007318(n-1,k-1), n > 0 and 0 <= k < n;
T(n,k) = T(n-1,k-2) + T(n-1,k), n > 0 and 2 <= k <= n-2;
T(n,2*k) = T(n-1,2*k) + T(n-1,2*(k-1)), k = 0..n;
T(n,2*k+1) = T(n-2,2*k), k = 0..n-1;
T(n,n) = A128014(n);
A105321(n) = number of odd terms in row n;
A249304(n) = number of even terms in row n;
T(n,k) mod 2 = A249133(n,k).

			The triangle begins:
.  0:                              1
.  1:                           1  1   1
.  2:                       1   1  2   1  1
.  3:                    1  1   3  2   3  1  1
.  4:                 1  1  4   3  6   3  4  1  1
.  5:              1  1  5  4  10  6  10  4  5  1  1
.  6:           1  1  6  5 15  10 20  10 15  5  6  1  1
.  7:        1  1  7  6 21 15  35 20  35 15 21  6  7  1  1
.  8:     1  1  8  7 28 21 56  35 70  35 56 21 28  7  8  1  1
.  9:  1  1  9  8 36 28 84 56 126 70 126 56 84 28 36  8  9  1  1 .

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A005408 (row lengths), A128014 (central terms), A003945 (row sums), A249111 (partial sums per row), A007318 (Pascal).

Cf. A105321, A249304, A249307, A249133.

import Data.List (transpose)
a249095 n k = a249095_tabf !! n !! k
a249095_row n = a249095_tabf !! n
a249095_tabf = [1] : map (concat . transpose)
   (zipWith ((. return) . (:)) (tail a007318_tabl) a007318_tabl)

t[n_, k_] := If[n > 1 && 1 < k < 2*n - 1, If[EvenQ[k], t[n - 1, k] + t[n - 1, k - 2], t[n - 1, k - 1]], 1]; Grid[Table[t[n, k], {n, 0, 9}, {k, 0, 2*n}]] (* L. Edson Jeffery, Nov 30 2014 *)

T(n,2*k) = T(n,2*k-1) + T(n,2*k+1), 0 < k < n.

A045680 Number of 2n-bead balanced binary necklaces of fundamental period 2n equivalent to reverse.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 17, 19, 64, 68, 245, 251, 900, 923, 3411, 3425, 12800, 12869, 48532, 48619, 184500, 184735, 705179, 705431, 2703168, 2704150, 10399675, 10400530, 40113164, 40116599, 155113825, 155117519, 601067520, 601080137, 2333593349

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
Index entries for sequences related to Lyndon words

Cf. A128014.

a[0] = 1;
a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[# - Mod[#, 2], Quotient[#, 2]]&];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

a(n) = if(n<1, n>=0, sumdiv(n, d, moebius(n/d) * binomial(d-d%2, d\2))); \\ Andrew Howroyd, Sep 28 2017

Moebius transform of A128014. - Andrew Howroyd, Sep 28 2017

A045675 Number of 2n-bead balanced binary necklaces which are not equivalent to their reverse, complement or reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 32, 168, 616, 2380, 8472, 30760, 109644, 394816, 1420784, 5149948, 18736744, 68553728, 251902032, 929814984, 3445433608, 12814382620, 47817551136, 178982546512, 671813695340, 2528191984504, 9536849826816

Offset: 0

David W. Wilson

nonn

The number of 2n-bead balanced binary necklaces is A003239(n). The number which are equivalent to their reverse, complement and reversed complement are respectively A128014(n), A000013(n) and A011782(n). - Andrew Howroyd, Sep 28 2017

Jean-François Alcover, Table of n, a(n) for n = 0..100
Index entries for sequences related to necklaces

Cf. A000013, A003239, A011782, A045674, A128014.

a3239[n_] := If[n==0, 1, Sum[EulerPhi[n/k]*Binomial[2k, k]/(2n), {k, Divisors[n]}]];
a128014[n_] := SeriesCoefficient[(1 + x)/Sqrt[1 - 4 x^2], {x, 0, n}];
a11782[n_] := SeriesCoefficient[(1 - x)/(1 - 2x), {x, 0, n}];
a13[n_] := If[n==0, 1, Sum[(EulerPhi[2d]*2^(n/d)), {d, Divisors[n]}]/(2n)];
a45674[n_] := a45674[n] = If[n==0, 1, If[EvenQ[n], 2^(n/2-1) + a45674[n/2], 2^((n-1)/2)]];
a[n_] := a3239[n] - a128014[n] - a13[n] - a11782[n] + 2 a45674[n];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 23 2019 *)

a(n) = A003239(n) - A128014(n) - A000013(n) - A011782(n) + 2*A045674(n). - Andrew Howroyd, Sep 28 2017

A128017 Expansion of (1+2x+x^2-x^3)/(1-x^2+x^4).

Original entry on oeis.org

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

Offset: 0

Paul Barry, Feb 11 2007

Hankel transform of A128014, A128015, A005717(n+1).

Index entries for linear recurrences with constant coefficients, signature (0,1,0,-1).

Cf. A128016.

A128055 a(n) = 2^A128054(n).

Original entry on oeis.org

1, 1, 2, 4, 8, 32, 64, 64, 128, 256, 512, 2048, 4096, 4096, 8192, 16384, 32768, 131072, 262144, 262144, 524288, 1048576, 2097152, 8388608, 16777216, 16777216, 33554432, 67108864, 134217728, 536870912, 1073741824

Offset: 0

Paul Barry, Feb 13 2007

A factor in A128056.

The signed sequence 1,1,2,-4,-8,-32,-64,-64,-128,256,512... is the Hankel transform of the doubled sequence 1,1,2,2,6,6,... of central binomial coefficients (A128014). - Paul Barry, Sep 09 2008

Index entries for linear recurrences with constant coefficients, signature (2,-4,8,-16,32).

Cf. A128014, A128054, A128056.

LinearRecurrence[{2,-4,8,-16,32},{1,1,2,4,8},31] (* James C. McMahon, Jan 05 2025 *)

G.f.: (-1-16*x^4+4*x^3-4*x^2+x)/((2*x-1)*(4*x^2-2*x+1)*(4*x^2+2*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

Sum_{n>=0} 1/a(n) = 62/21. - Amiram Eldar, Mar 28 2023

A249308 Central terms of triangle A249307.

Original entry on oeis.org

1, 2, 8, 16, 96, 192, 1280, 2560, 17920, 35840, 258048, 516096, 3784704, 7569408, 56229888, 112459776, 843448320, 1686896640, 12745441280, 25490882560, 193730707456, 387461414912, 2958796259328, 5917592518656, 45368209309696, 90736418619392, 697972450918400

Offset: 0

Reinhard Zumkeller, Nov 14 2014

nonn

a(n) = A128014(n) * 2^n = A249307(n,n);

A098430(n) = a(2*n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A249307, A128014, A098430.

Haskell
```
a249308 n = a249307 n n
```

A045676 Number of 2n-bead balanced binary necklaces which are equivalent to their reverse, but not equivalent to their complement and reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 14, 12, 58, 54, 232, 220, 886, 860, 3360, 3304, 12730, 12614, 48348, 48108, 184224, 183732, 704376, 703384, 2702070, 2700060, 10396440, 10392408, 40108336, 40100216, 155101008, 155084752, 601047482, 601014854, 2333540428

Offset: 0

David W. Wilson

nonn

The number of 2n-bead balanced binary necklaces equivalent to their reverse is A128014(n) and those equivalent to their reverse, complement and reversed complement is A045674(n). - Andrew Howroyd, Sep 28 2017

Index entries for sequences related to necklaces

Cf. A045674, A045675, A045677, A045678, A128014.

A045674[n_] := A045674[n] = If[n == 0, 1, If[EvenQ[n], 2^(n/2 - 1) + A045674[n/2], 2^((n-1)/2)]];
a[n_] := SeriesCoefficient[(1+x)/Sqrt[1 - 4 x^2], {x, 0, n}] - A045674[n];
a /@ Range[0, 34] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = A128014(n) - A045674(n). - Andrew Howroyd, Sep 28 2017

A128057 Expansion of (1+x)/sqrt(1+4x^2).

Original entry on oeis.org

1, 1, -2, -2, 6, 6, -20, -20, 70, 70, -252, -252, 924, 924, -3432, -3432, 12870, 12870, -48620, -48620, 184756, 184756, -705432, -705432, 2704156, 2704156, -10400600, -10400600, 40116600, 40116600, -155117520, -155117520

Offset: 0

Paul Barry, Feb 13 2007

Hankel transform is A128056. Binomial transform is A128058. Unsigned version is A128014.

Cf. A000984.

CoefficientList[Series[(1+x)/Sqrt[1+4x^2],{x,0,40}],x] (* Harvey P. Dale, May 12 2015 *)

a(n)=(-1)^C(n,2)*(C(n,n/2)*(1+(-1)^n)/2+C(n-1,(n-1)/2)*(1-(-1)^n)/2);
Conjecture: n*(5*n-9)*a(n) +4*a(n-1) +4*(5*n-4)*(n-2)*a(n-2)=0. - R. J. Mathar, Dec 02 2014
D-finite with recurrence: n*a(n) +(n-2)*a(n-1) +4*(n-1)*a(n-2) +4*(n-3)*a(n-3)=0. - R. J. Mathar, Dec 02 2014

cp's OEIS Frontend

A126869 a(n) = Sum_{k = 0..n} binomial(n,floor(k/2))*(-1)^(n-k).

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A128018 Expansion of (1-4*x)/(1-2*x+4*x^2).

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A249095 Triangle read by rows: interleaving successive pairs of rows of Pascal's triangle.

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A045680 Number of 2n-bead balanced binary necklaces of fundamental period 2n equivalent to reverse.

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A045675 Number of 2n-bead balanced binary necklaces which are not equivalent to their reverse, complement or reversed complement.

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A128017 Expansion of (1+2x+x^2-x^3)/(1-x^2+x^4).

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A128055 a(n) = 2^A128054(n).

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A249308 Central terms of triangle A249307.

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A128018 Expansion of (1-4x)/(1-2x+4*x^2).