A181983 -id:A181983 - cp's OEIS frontend

A270143 a(n) = Sum_{k=0..n} (-1)^(k+1) * k * A000041(n-k).

0, 1, -1, 3, -2, 6, -3, 11, -4, 19, -4, 31, -2, 50, 3, 79, 15, 122, 38, 187, 78, 284, 146, 426, 257, 635, 431, 939, 701, 1377, 1110, 2007, 1718, 2906, 2613, 4178, 3914, 5971, 5781, 8482, 8440, 11976, 12191, 16816, 17438, 23483, 24730, 32615, 34794, 45070

Offset: 0

Vaclav Kotesovec, Mar 12 2016

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Convolution of A000041 and A181983.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000041, A014153, A087787, A270144.

Table[Sum[(-1)^(n-k+1)*PartitionsP[k]*(n-k), {k, 0, n}], {n, 0, 100}]
nmax = 100; CoefficientList[Series[x/(1 + x)^2 * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = Sum_{k=0..n} (-1)^(n-k+1) * (n-k) * A000041(k).
a(n) ~ A000041(n)/4.
a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)).
G.f.: x/(1+x)^2 * Product_{k>=1} 1/(1-x^k).

A270144 a(n) = Sum_{k=0..n} (-1)^(k+1) * k * A000009(n-k).

Original entry on oeis.org

0, 1, -1, 2, -1, 2, 0, 2, 1, 2, 3, 2, 5, 3, 7, 5, 10, 7, 14, 11, 18, 17, 24, 24, 32, 34, 42, 47, 56, 63, 74, 85, 96, 113, 126, 147, 165, 191, 213, 247, 275, 316, 353, 404, 449, 514, 571, 648, 723, 816, 909, 1024, 1140, 1278, 1424, 1592, 1770, 1976, 2195, 2442

Offset: 0

Vaclav Kotesovec, Mar 12 2016

sign

Convolution of A000009 and A181983.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000009, A025147, A095944, A270143.

Table[Sum[(-1)^(n-k+1)*PartitionsQ[k]*(n-k), {k, 0, n}], {n, 0, 100}]
nmax = 100; CoefficientList[Series[x/(1 + x)^2 * Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = Sum_{k=0..n} (-1)^(n-k+1) * (n-k) * A000009(k).
a(n) ~ A000009(n)/4.
a(n) ~ exp(Pi*sqrt(n/3)) / (16*3^(1/4)*n^(3/4)).
G.f.: x/(1+x)^2 * Product_{k>=1} (1+x^k).

A274703 Exponential generating function 1/M_{3}(z^3) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only.

Original entry on oeis.org

1, -4, 133, -15130, 4101799, -2177360656, 1999963458217, -2919514870785766, 6365117686550339275, -19765974970578036695068, 84220118333781814726917709, -477722110504065444764182065202, 3518554409906597166261453268226671, -32952557456293494405944914420304822440

Offset: 0

Peter Luschny, Jul 03 2016

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For references see also A274705 which is the main entry for this sequence of sequences.

Robert Israel, Table of n, a(n) for n = 0..166
Eric Weisstein's MathWorld, Mittag-Leffler Function
Wikipedia, Mittag-Leffler function

Cf. A181983 (n=1), A009843 (n=2), A274704 (n=4), A274705 (array).

s := series(z/((exp(z)+2*exp(-z/2)*cos(z*3^(1/2)/2))/3),z,60):
seq((n*3+1)!*coeff(s,z,n*3+1), n=0..13);

c = CoefficientList[Series[1/MittagLefflerE[3,z^3],{z,0,15*3}],z];
Table[Factorial[3*n+1]*c[[3*n+1]], {n,0,13}]

E.g.f. (nonzero coefficients): z/((exp(z)+2*exp(-z/2)*cos(z*3^(1/2)/2))/3).

For n >= 1, a(n) = -Sum_{k=0..n-1} a(k) binomial(3n+1,3k+1). - Robert Israel, Jul 03 2016

A274704 Exponential generating function 1/M_{4}(z^4) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only.

Original entry on oeis.org

1, -5, 621, -437593, 1026405753, -6054175060941, 75477454065058725, -1766732850877953050849, 71248914440011028226682737, -4637564239713542128355021380117, 462852368857623061805761137170608989, -67965094887205237792816627191801312013545

Offset: 0

Peter Luschny, Jul 03 2016

sign

Robert Israel, Table of n, a(n) for n = 0..128

Cf. A181983 (n=1), A009843 (n=2), A274703 (n=3), A274705 (array).

s := series(2*z/(cosh(z)+cos(z)),z,60):
seq((4*n+1)!*coeff(s,z,4*n+1),n=0..11);

c = CoefficientList[Series[1/MittagLefflerE[4, z^4], {z, 0, 15*4}], z];
Table[Factorial[4*n+1]*c[[4*n+1]], {n, 0, 12}]

E.g.f. (nonzero coefficients): 2*z/(cosh(z)+cos(z)).

For n >= 1, a(n) = - Sum_{k=0..n-1} a(k)*binomial(4*k+1,4*n+1). - Robert Israel, Jul 04 2016

A374157 a(n) = (-1)^floor(n/2)*n.

Original entry on oeis.org

0, 1, -2, -3, 4, 5, -6, -7, 8, 9, -10, -11, 12, 13, -14, -15, 16, 17, -18, -19, 20, 21, -22, -23, 24, 25, -26, -27, 28, 29, -30, -31, 32, 33, -34, -35, 36, 37, -38, -39, 40, 41, -42, -43, 44, 45, -46, -47, 48, 49, -50, -51, 52, 53, -54, -55, 56, 57, -58, -59

Offset: 0

Peter Luschny, Jun 30 2024

For all odd numbers n (A005408) and all whole numbers z (A001057) K(z/n) = K(a(n)/z), where K(z/n) denotes the Kronecker symbol (A372728). This fact is equivalent to the law of quadratic reciprocity and its first and second supplement.

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A005408, A001057, A057077, A196521, A372728.

Cf. A001477, A181983.

a := n -> (-1)^iquo(n, 2)*n: seq(a(n), n = 0..59);

Array[(-1)^Floor[#/2]*# &, 60, 0] (* Michael De Vlieger, Jun 30 2024 *)

a(n) = (-1)^(n\2) * n; \\ Amiram Eldar, Jun 30 2024

def A374157(n): return (-1)**(n // 2)*n

def A374157(n): return -n if n&2 else n # Chai Wah Wu, Jun 30 2024

Sum_{n>=1} 1/a(n) = Pi/4 - log(2)/2 = A196521.
a(n) = [x^n] -x*(x^2 + 2*x - 1)/(x^2 + 1)^2.
a(n) = n! * [x^n] x*(cos(x) - sin(x)). - Stefano Spezia, Jun 30 2024
a(n) = n*A057077(n). - Michel Marcus, Jul 01 2024

A386789 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).

Original entry on oeis.org

1, 0, 2, 0, 3, 6, 0, 4, 20, 20, 0, 5, 45, 105, 70, 0, 6, 84, 336, 504, 252, 0, 7, 140, 840, 2100, 2310, 924, 0, 8, 216, 1800, 6600, 11880, 10296, 3432, 0, 9, 315, 3465, 17325, 45045, 63063, 45045, 12870, 0, 10, 440, 6160, 40040, 140140, 280280, 320320, 194480, 48620

Offset: 0

Peter Luschny, Aug 06 2025

			Triangle begins:
  [0] 1;
  [1] 0, 2;
  [2] 0, 3,   6;
  [3] 0, 4,  20,   20;
  [4] 0, 5,  45,  105,   70;
  [5] 0, 6,  84,  336,  504,   252;
  [6] 0, 7, 140,  840, 2100,  2310,   924;
  [7] 0, 8, 216, 1800, 6600, 11880, 10296, 3432;
.
Seen as an array A(n, k) = binomial(n + k - 1, n)*binomial(n + 2*k, k):
  [0] 1, 2,   6,   20,    70,    252,     924, ... [A000984]
  [1] 0, 3,  20,  105,   504,   2310,   10296, ... [A000917]
  [2] 0, 4,  45,  336,  2100,  11880,   63063, ...
  [3] 0, 5,  84,  840,  6600,  45045,  280280, ...
  [4] 0, 6, 140, 1800, 17325, 140140, 1009008, ...
  [5] 0, 7, 216, 3465, 40040, 378378, 3118752, ...
  [6] 0, 8, 315, 6160, 84084, 917280, 8576568, ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Cf. A176479 (row sums), A000984 (main diagonal), A181983 (alternating row sums), A386876 (central terms).

Cf. A000984, A000917, A357367.

T := (n, k) -> binomial(n - 1, k - 1)*binomial(n + k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);

A386789[n_, k_] := Binomial[n - 1, k - 1]*Binomial[n + k, k];
Table[A386789[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 06 2025 *)

A357367(n, k) = n!*T(n, k).

A030640 Discriminant of lattice A_n of determinant n+1.

Original entry on oeis.org

1, 1, -3, -2, 5, 3, -7, -4, 9, 5, -11, -6, 13, 7, -15, -8, 17, 9, -19, -10, 21, 11, -23, -12, 25, 13, -27, -14, 29, 15, -31, -16, 33, 17, -35, -18, 37, 19, -39, -20, 41, 21, -43, -22, 45, 23, -47, -24, 49, 25, -51, -26, 53, 27, -55, -28, 57, 29, -59

Offset: 0

N. J. A. Sloane

			G.f. = 1 + x - 3*x^2 - 2*x^3 + 5*x^4 + 3*x^5 - 7*x^6 - 4*x^7 + 8*x^9 + 5*x^10 + ...

J. H. Conway, The Sensual Quadratic Form, Mathematical Association of America, 1997, p. 4.
G. L. Watson, Integral Quadratic Forms, Cambridge University Press, p. 2.

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A026741 is unsigned version.

Cf. A157142, A181983.

CoefficientList[Series[(1+x-x^2)/(1+x^2)^2,{x,0,60}],x] (* or *) LinearRecurrence[{0,-2,0,-1},{1,1,-3,-2},70]
a[ n_] := With[{m = n + 1}, m I^m / If[ Mod[ m, 2] == 1, I, -2]]; (* Michael Somos, Jun 11 2013 *)

{a(n) = if( n==-1, 0, (-1)^(n\2) * (n+1) / gcd(n+1, 2))}; /* Michael Somos, Jun 15 2005 */

def A030640(n): return (-(n+1>>1) if n&2 else n+1>>1) if n&1 else (-n-1 if n&2 else n+1) # Chai Wah Wu, Aug 05 2024

a(2n) = (-1)^n*(2*n+1), a(2n+1) = (-1)^n*(n+1). Or (apart from signs and with offset 1), a(n) = n, n odd; n/2, n even.
G.f.: (1+x-x^2)/(1+x^2)^2. - Len Smiley
a(-2-n) = (-1)^n * a(n). - Michael Somos, Jun 15 2005
a(n) = -2*a(n-2) - a(n-4); a(0)=1, a(1)=1, a(2)=-3, a(3)=-2. - Harvey P. Dale, Dec 02 2011
a(n) = (-1)^floor(n/2)*A026741(n+1).
a(2*n) = A157142(n). a(2*n - 1) = A181983(n). - Michael Somos, Feb 22 2016

A131738 a(0) = 0. a(n) = (n+1)*(-1)^n, n>0 .

Original entry on oeis.org

0, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -32, 33, -34, 35, -36, 37, -38, 39, -40, 41, -42, 43, -44, 45, -46, 47, -48, 49, -50, 51, -52, 53, -54, 55, -56, 57, -58, 59, -60, 61, -62, 63, -64, 65, -66, 67, -68, 69, -70, 71, -72, 73, -74, 75

Offset: 0

Paul Curtz, Sep 19 2007

Also the main diagonal of A138057.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A105811.

Cf. A181983 (main entry).

[0] cat [(-1)^n*(n+1): n in [1..50]]; // G. C. Greubel, Nov 02 2017

Join[{0}, Table[(-1)^n*(n + 1), {n, 1, 50}]] (* G. C. Greubel, Nov 02 2017 *)
LinearRecurrence[{-2,-1},{0,-2,3},80] (* Harvey P. Dale, Dec 03 2022 *)

a(n)=if(n,(n+1)*(-1)^n,0) \\ Charles R Greathouse IV, Sep 01 2015

From G. C. Greubel, Nov 02 2017: (Start)
a(n) = -2*a(n-1) - a(n-2).
G.f.: -x*(x+2)/(1+x)^2.
E.g.f.: (1 - x - exp(x))*exp(-x). (End)

Edited by R. J. Mathar, Jul 07 2008

A274705 Rectangular array read by ascending antidiagonals. Row n has the exponential generating function 1/M_{n}(z^n) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only, for n>=1.

Original entry on oeis.org

1, 1, -2, 1, -3, 3, 1, -4, 25, -4, 1, -5, 133, -427, 5, 1, -6, 621, -15130, 12465, -6, 1, -7, 2761, -437593, 4101799, -555731, 7, 1, -8, 11999, -12012016, 1026405753, -2177360656, 35135945, -8, 1, -9, 51465, -325204171, 243458990271, -6054175060941, 1999963458217, -2990414715, 9

Offset: 0

Peter Luschny, Jul 03 2016

			Array starts:
n=1: {1, -2,  3, -4, 5, -6, 7, -8,  9, -10,  11,...} [A181983]
n=2: {1, -3, 25, -427, 12465, -555731, 35135945,...} [A009843]
n=3: {1, -4, 133, -15130, 4101799,  -2177360656,...} [A274703]
n=4: {1, -5, 621, -437593, 1026405753, -6054175060941,...} [A274704]
n=5: {1, -6, 2761, -12012016, 243458990271, ...}

L. Carlitz, Some arithmetic properties of the Olivier functions, Math. Ann. 128 (1954), 412-419.
H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics, vol. 2011, Article ID 298628, 51 pages.
L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
Eric Weisstein's MathWorld, Generalized hyperbolic functions.

Cf. A009843, A181983, A274703, A274704.

ibn := proc(m, k) local w, om, t;
w := exp(2*Pi*I/m); om := m*x/add(exp(x*w^j), j=0..m-1);
t := series(om, x, k+m); simplify(k!*coeff(t,x,k)) end:
seq(seq(ibn(n-k+2, n*k-n-k^2+3*k-1), k=1..n+1),n=0..8);

A274705Row[m_] := Module[{c}, c = CoefficientList[Series[1/MittagLefflerE[m,z^m],
{z,0,12*m}],z]; Table[Factorial[m*n+1]*c[[m*n+1]], {n,0,9}] ]
Table[Print[A274705Row[n]], {n,1,6}]

def ibn(m, k):
    w = exp(2*pi*I/m)
    om = m*x/sum(exp(x*w^j) for j in range(m))
    t = taylor(om, x, 0, k + m)
    return simplify(factorial(k)*t.list()[k])
def A274705_row(m, size):
    return [ibn(m, k) for k in range(1, m*size, m)]
for n in (1..4): print(A274705_row(n, 8))

Recurrence for the m-th row: R(m, n) = -Sum_{k=0..n-1} binomial(m*n+1, m*k+1)*R(m, k) for n >= 1. See Carlitz (1.3).

A324221 Number of connected 2n-regular loopless multigraphs with five nodes.

Original entry on oeis.org

0, 1, 6, 15, 36, 72, 139, 244, 414, 663, 1030, 1540, 2247, 3187, 4433, 6036, 8088, 10658, 13861, 17785, 22571, 28329, 35227, 43401, 53049, 64333, 77485, 92697, 110235, 130324, 153268, 179326, 208843, 242115, 279529, 321422, 368226, 420319, 478182, 542238, 613017

Offset: 0

Natan Arie Consigli, Feb 18 2019

nonn

There are no (2n+1)-regular multigraphs satisfying the condition above.
Multigraphs are loopless.
Initial terms computed with 'Nauty and Traces'.

Brendan McKay and Adolfo Piperno, Nauty and Traces

Row n=5 of A328682.

for ((n=0;n<76;n=n+2)); do geng -c -d1 5 -q | multig -m${n} -u; done

Conjectures from Colin Barker, Feb 18 2019: (Start)
G.f.: x*(1 + 3*x - x^2 + 4*x^3 - x^4 + 6*x^5 + 4*x^7 - x^8 - x^9 + x^10) / ((1 - x)^6*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) - 2*a(n-5) + 4*a(n-6) - 2*a(n-7) + a(n-8) - a(n-9) - 2*a(n-10) + 3*a(n-11) - a(n-12) for n>11.
(End)
Equivalent conjecture: 1152*a(n) = 6*n^5 + 30*n^4 + 220*n^3 + 540*n^2 + 1143*n - 353 + 72*A056594(n) + 128*A049347(n) + 153*A181983(n+1). - R. J. Mathar, Mar 09 2019

a(28)-a(30) from Andrew Howroyd, Mar 18 2020

cp's OEIS Frontend

A270143 a(n) = Sum_{k=0..n} (-1)^(k+1) * k * A000041(n-k).

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A270144 a(n) = Sum_{k=0..n} (-1)^(k+1) * k * A000009(n-k).

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A274703 Exponential generating function 1/M_{3}(z^3) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only.

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A274704 Exponential generating function 1/M_{4}(z^4) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only.

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A374157 a(n) = (-1)^floor(n/2)*n.

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A386789 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).

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A030640 Discriminant of lattice A_n of determinant n+1.

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