A009116 -id:A009116 - cp's OEIS frontend

A057681 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j).

1, 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, 129140163, 258280326, 387420489, 387420489

Offset: 0

N. J. A. Sloane, Oct 20 2000

Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1-M)^n = a(n)-A057682(n)*M+z(n)*M^2, where z(0)=z(1)=0 and, apparently, z(n+2)=A057083(n). - Stanislav Sykora, Jun 10 2012
Pisano period lengths: 1, 3, 1, 6, 24, 3, 6, 12, 1, 24, 60, 6, 12, 6, 24, 24, 96, 3, 18, 24, ... . - R. J. Mathar, Aug 10 2012
{A057681, A057682, A*}, where A* is A057083 prefixed by two 0's, is the difference analog of the trigonometric functions of order 3, {k_1(x), k_2(x), k_3(x)}. For a definition see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jun 25 2017

			If M^3=1 then (1-M)^6 = a(6)-A057682(6)*M+A057083(4)*M^2 = -18+9*M+9*M^2.

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
T. Alden Gassert, Discriminants of simplest 3^n-tic extensions, arXiv:1409.7829 [math.NT], 2014.
Mark W. Coffey, Reductions of particular hypergeometric functions 3F2 (a, a+1/3, a+2/3; p/3, q/3; +-1), arXiv:1506.09160 [math.CA], 2015.
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv:1610.09361 [math.NT], 2016.
Ira Gessel, The Smith College diploma problem.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3).

Cf. A009116, A009545, A057682, A057083, A103312.

a:=[1,1];; for n in [3..40] do a[n]:=3*a[n-1]-3*a[n-2]; od; Concatenation([1],a); # Muniru A Asiru, Oct 24 2018

I:=[1,1]; [1] cat [n le 2 select I[n] else 3*Self(n-1) - 3*Self(n-2): n in [1..40]]; // G. C. Greubel, Oct 23 2018

A057681 := n->add((-1)^j*binomial(n,3*j),j=0..floor(n/3)); seq(A057681(n), n=0..50);
A057681_list := proc(n) local i; series((1+2*exp(3*z/2)*cos(z*sqrt(3/4)))/3, z,n+2): seq(i!*coeff(%,z,i),i=0..n) end: A057681_list(38); # Peter Luschny, Jul 10 2012

Join[{1},LinearRecurrence[{3,-3},{1,1},40]] (* Harvey P. Dale, Aug 19 2014 *)

x='x+O('x^40); Vec((1-x)^2/((1-x)^3+x^3)) \\ G. C. Greubel, Oct 23 2018

From Paul Barry, Feb 26 2004: (Start)
G.f.: (1-x)^2/((1-x)^3+x^3).
a(n) = 0^n/3 + 2*3^((n-2)/2)*cos(Pi*n/6). (End)
From Paul Barry, Feb 27 2004: (Start)
Binomial transform of (1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, ...).
E.g.f.: 2*exp(3x/2)*cos(sqrt(3)*x/2)/3+1/3.
a(n) = (((3+sqrt(-3))/2)^n+((3-sqrt(-3))/2)^n)/3+0^n/3. (End)
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5). - Paul Curtz, Jan 02 2008
Start with x(0)=1,y(0)=0,z(0)=0 and set x(n+1)=x(n)-z(n), y(n+1)=y(n)-x(n),z(n+1)=z(n)-y(n). Then a(n)=x(n). But this recurrence falls into a repetitive cycle of length 6 and multiplicative factor -27, so that a(n) = -27*a(n-6) for any n>6. - Stanislav Sykora, Jun 10 2012
E.g.f.: (1+2*exp(3*z/2)*cos(z*sqrt(3/4)))/3. - Peter Luschny, Jul 10 2012
a(0)=a(1)=a(2)=1, a(n)=3*a(n-1)-3*a(n-2), n>=3. - Wesley Ivan Hurt, Nov 11 2014
For n>=1, a(n) = 2*3^((n-2)/2)*cos(Pi*n/6). - Vladimir Shevelev, Jun 25 2017
a(n+m) = a(n)*a(m)-A057682(n)*A*057083(m)-A*057083(n)*A057682(m), where A*057083 is A057083 prefixed by two 0's. - Vladimir Shevelev, Jun 25 2017

A104597 Triangle T read by rows: inverse of Motzkin triangle A097609.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -1, -2, 0, 1, 0, -2, -3, 0, 1, 1, 1, -3, -4, 0, 1, 1, 4, 3, -4, -5, 0, 1, 0, 3, 9, 6, -5, -6, 0, 1, -1, -2, 5, 16, 10, -6, -7, 0, 1, -1, -6, -9, 6, 25, 15, -7, -8, 0, 1, 0, -4, -18, -24, 5, 36, 21, -8, -9, 0, 1, 1, 3, -7, -39, -50, 1, 49, 28, -9, -10, 0, 1, 1, 8

Offset: 0

Ralf Stephan, Mar 17 2005

Riordan array ((1-x)/(1-x+x^2),x(1-x)/(1-x+x^2)). - Paul Barry, Jun 21 2008

			1
0,1
-1,0,1
-1,-2,0,1
0,-2,-3,0,1
1,1,-3,-4,0,1
1,4,3,-4,-5,0,1
0,3,9,6,-5,-6,0,1
-1,-2,5,16,10,-6,-7,0,1
-1,-6,-9,6,25,15,-7,-8,0,1

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [alternative link]
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.

Row sums are A009116 with different signs.
Row sums are A146559(n).
Cf. A005043, A091867, A030528, A000108.

# Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> A005043(n-1)); # Peter Luschny, Oct 09 2022

T(n,m):=sum(binomial(m,j)*sum(binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1),k,0,n)*(-1)^(m-j),j,0,m); /* Vladimir Kruchinin, Apr 08 2011 */

T(n,m) = sum(j=0..m, binomial(m,j)*sum(k=0..n, binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1))*(-1)^(m-j)). - Vladimir Kruchinin, Apr 08 2011
T(n,m) = sum(k=ceiling((n-m-1)/2)..n-m, binomial(k+m,m)*binomial(k+1,n-k-m)*(-1)^(n-k-m)). - Vladimir Kruchinin, Dec 17 2011
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 20 2013
T(n+5,n) = (n+1)^2. - Philippe Deléham, Feb 20 2013
From Tom Copeland, Nov 04 2014: (Start)
O.g.f.: G(x,t) = Pinv[Cinv(x),t+1] = Cinv(x) / [1 - (t+1)Cinv(x)] = x*(1-x) / [1-(t+1)x(1-x)] = x + t * x^2 + (-1 + t^2) * x^3 + ..., where Cinv(x)= x * (1-x) is the inverse of C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the Catalan numbers A000108 and Pinv(x,t) = -P(-x,t) = x/(1-t*x) is the inverse of P(x,t) = x/(1+x*t).
Ginv(x,t)= C[P[x,t+1]]= C[x/(1+(t+1)x)] = {1-sqrt[1-4*x/(1+(t+1)x)]}/2.
The inverse in x of G(x,t) with t replaced by -t is the o.g.f. of A091867, and G(x,t-1) is a signed version of the (mirrored) Fibonacci polynomials A030528. (End)

A084102 Inverse binomial transform of A084101.

Original entry on oeis.org

1, 2, -2, 0, 4, -8, 8, 0, -16, 32, -32, 0, 64, -128, 128, 0, -256, 512, -512, 0, 1024, -2048, 2048, 0, -4096, 8192, -8192, 0, 16384, -32768, 32768, 0, -65536, 131072, -131072, 0, 262144, -524288, 524288, 0, -1048576, 2097152, -2097152, 0, 4194304, -8388608, 8388608, 0, -16777216, 33554432

Offset: 0

Paul Barry, May 15 2003

The sequence {2, -2, 0, 4, -8, 8, 0, -16, 32, -32, 0, 64, -128, 128, 0, ...} (without the leading 1) is the Lucas V(-2, 2) sequence. - R. J. Mathar, Jan 08 2013

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

Yet another variation on A009545.

Cf. A009116, A084101.

[1] cat [n le 2 select 2*(-1)^(n-1) else -2*(Self(n-1) +Self(n-2)): n in [1..40]]; // G. C. Greubel, Oct 13 2022

LinearRecurrence[{-2,-2},{1,2,-2},50] (* Harvey P. Dale, Aug 09 2017 *)

b=BinaryRecurrenceSequence(-2,-2,2,-2)
def A084102(n): return 1 if (n==0) else b(n-1)
[A084102(n) for n in range(41)] # G. C. Greubel, Oct 13 2022

G.f.: (1+2*x)^2/(1+2*x+2*x^2). - Paul D. Hanna, Nov 05 2009
From G. C. Greubel, Oct 13 2022: (Start)
a(n) = 2*A009116(n-1), n >= 1, with a(0) = 1.
a(n) = Real part of ( 2*(-1-i)^(n-1) + 2*[n=0] ).
a(n) = 2*(-1)^n*(2*(1+i)^(n-5) + i*(1-i)^(n-3)), n >= 1, with a(0) = 1.
E.g.f.: 2 - exp(-x)*(cos(x) - sin(x)). (End)

A090132 Expansion of (1+2x)/(1+2x+2*x^2).

Original entry on oeis.org

1, 0, -2, 4, -4, 0, 8, -16, 16, 0, -32, 64, -64, 0, 128, -256, 256, 0, -512, 1024, -1024, 0, 2048, -4096, 4096, 0, -8192, 16384, -16384, 0, 32768, -65536, 65536, 0, -131072, 262144, -262144, 0, 524288, -1048576, 1048576, 0, -2097152, 4194304, -4194304, 0, 8388608

Offset: 0

Paul Barry, Nov 21 2003

The expansion of (1-2x)/(1-2x+2x^2) has a(n) = Sum_{k=0..n} C(n,k)(-1)^(-k)(-1)^floor(k/2).

Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012

			G.f. = 1 - 2*x^2 + 4*x^3 - 4*x^4 + 8*x^6 - 16*x^7 + 16*x^8 - 32*x^10 + 64*x^11 + ...

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

Cf. A009116.

Cf. A135353, A137444, A137426, A137429.

a[ n_] := Re[ -(I - 1)^(n + 1)]; (* Michael Somos, May 25 2013 *)
a[ n_] := If[ n < 0, - 2^(n-1) a[2 - n], SeriesCoefficient[ (1 + 2 x) / (1 + 2 x + 2 x^2), {x, 0, n}]]; (* Michael Somos, May 25 2013 *)
a[ n_] := If[ n < 0, - 2^(n-1) a[2 - n], n! SeriesCoefficient[ (Cos[x] + Sin[x]) / Exp[x], {x, 0, n}]]; (* Michael Somos, May 25 2013 *)
a[ n_] := Simplify[ -2 Sqrt[2]^(n - 1) ChebyshevT[ n + 1, -1 / Sqrt[2]]]; (* Michael Somos, May 25 2013 *)
LinearRecurrence[{-2,-2},{1,0},50] (* Harvey P. Dale, Oct 23 2017 *)

my(x='x+O('x^66)); Vec(serlaplace((cos(x)+sin(x))/exp(x))) \\ Joerg Arndt, May 13 2011

vector(66, n, -real((-1+I)^n)) /* Joerg Arndt, May 13 2011 */

{a(n) = real( -(I - 1)^(n + 1) )}; /* Michael Somos, May 25 2013 */

{a(n) = if( n<0, - 2^(n-1) * a(2 - n), polcoeff( (1 + 2*x) / (1 + 2*x + 2*x^2) + x * O(x^n), n))}; /* Michael Somos, May 25 2013 */

{a(n) = my(A); if( n<0, - 2^(n-1) * a(2 - n), A = x * O(x^n); n! * polcoeff( (cos(x + A) + sin(x + A)) / exp(x + A), n))}; /* Michael Somos, May 25 2013 */

{a(n) = simplify( -2 * quadgen(8)^(n - 1) * polchebyshev( n + 1, 1, -1 / quadgen(8)))}; /* Michael Somos, May 25 2013 */

G.f.: (1+2*x)/(1+2*x+2*x^2).
a(n) = Sum_{k=0..n} C(n, k)*(-1)^(n-k)*(-1)^floor(k/2).
a(n) = sqrt(2)*2^(n/2)*sin(3*Pi*n/4+Pi/4). - Paul Barry, Feb 25 2004
a(n) = -a(n-1) + 2*a(n-3). - Paul Curtz, Apr 24 2008
Negated real part of (-1+i)^n, imaginary part is A108520. - Joerg Arndt, May 13 2011
From Sergei N. Gladkovskii, Nov 28 2011: (Start)
E.g.f.: (cos(x) + sin(x))/exp(x).
E.g.f.: A(x) = Q(0), where Q(k)=1-(x^2)/((4*k+1)*(2*k+1)+2*x*(4*k+1)*(2*k+1)/(4*k+3-2*x-x*(4*k+3)/(x-(4*k+4)/Q(k+1)))); (continued fraction). (End)
a(4*n + 1) = 0. a(2*n) = A120617(n). a(4*n + 3) = (-4)^n. - Michael Somos, May 25 2013
a(n) = - 2^(n-1)*a(2-n) for all n in Z. - Michael Somos, Jun 26 2017
a(n) = (I + 1)*((-1 - I)^n - I*(-1 + I)^n)/2. - Taras Goy, Apr 20 2019

A115416 Imaginary part of (n + i)^n, with i=sqrt(-1).

Original entry on oeis.org

0, 1, 4, 26, 240, 2876, 42372, 740536, 14970816, 343603216, 8825080100, 250756091552, 7809130867824, 264489160965056, 9678967816041188, 380574552503498624, 16000787866533953280, 716309568462681538816

Offset: 0

Reinhard Zumkeller, Jan 22 2006

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A000312, A009116, A115415 (real part).

seq(Im((n+I)^n), n=0..20); # Robert Israel, Dec 30 2024

Table[Im[(n + I)^n], {n, 0, 17}] (* Robert G. Wilson v, Jan 23 2006 *)

a(n) = imag((n + I)^n); \\ Michel Marcus, Apr 11 2018

from math import comb
def A115416(n): return sum(comb(n,j)*n**(n-j)*(-1 if j-1&2 else 1) for j in range(1,n+1,2)) # Chai Wah Wu, Feb 15 2024

a(n) = n! * [x^n] exp(n*x)*sin(x). - Ilya Gutkovskiy, Apr 10 2018
a(n) ~ sin(1) * n^n. - Vaclav Kotesovec, Jun 08 2019
a(n) = Sum_{j=0..floor((n-1)/2)} binomial(n,2*j+1)*n^(n-2*j-1)*(-1)^j. - Chai Wah Wu, Feb 15 2024
a(n) = -(i/2)*((n + i)^n - (n - i)^n) where i is the imaginary unit. - Gerry Martens, Dec 30 2024

More terms from Robert G. Wilson v, Jan 23 2006

A318438 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let h(n) = Sum_{k=0..w} b_k * (i-1)^k (where i denotes the imaginary unit); a(n) is the real part of h(n).

Original entry on oeis.org

0, 1, -1, 0, 0, 1, -1, 0, 2, 3, 1, 2, 2, 3, 1, 2, -4, -3, -5, -4, -4, -3, -5, -4, -2, -1, -3, -2, -2, -1, -3, -2, 4, 5, 3, 4, 4, 5, 3, 4, 6, 7, 5, 6, 6, 7, 5, 6, 0, 1, -1, 0, 0, 1, -1, 0, 2, 3, 1, 2, 2, 3, 1, 2, 0, 1, -1, 0, 0, 1, -1, 0, 2, 3, 1, 2, 2, 3, 1, 2

Offset: 0

Rémy Sigrist, Aug 26 2018

See A318439 for the imaginary part of h.
See A318479 for the square of the modulus of h.
The function h corresponds to the interpretation of the binary representation of a number in base -1+i and defines a bijection from the nonnegative integers to the Gaussian integers.
The function h has nice fractal features (see scatterplot in Links section).
This sequence has similarities with A316657.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of (a(n), A318439(n)) for n = 0..2^20-1 (where the hue is function of n)
Wikipedia, Base -1+/-i
Index entries for sequences related to binary expansion of n

Cf. A009116, A318439 (imaginary part), A318479 (norm), A340669 (negation).

Cf. A316657 (base 2+i).

a(n) = my (d=Vecrev(digits(n,2))); real(sum(i=1, #d, d[i]*(I-1)^(i-1)))

a(2^k) = A009116(k) for any k >= 0.

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A100213 Expansion of g.f.: x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x) (1-2x^2) (1-2x+2x^2) * (1+2*x^2)).

Original entry on oeis.org

4, 9, 14, 18, 32, 64, 128, 256, 544, 1104, 2144, 4128, 8192, 16384, 32768, 65536, 131584, 263424, 525824, 1049088, 2097152, 4194304, 8388608, 16777216, 33562624, 67129344, 134242304, 268443648, 536870912, 1073741824, 2147483648, 4294967296, 8590065664

Offset: 1

Creighton Dement, Nov 11 2004

The sequence can be created applying the pos operator (which sums over the positive coefficients) to the n-th power of the Floretion element (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e).

			a(5) = 32 because (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)^5 = - 2 'j - 2 'k - 2 j' - 2 k' + 6 'ii' + 10 'jj' + 10 'kk' + 6 e,
and the sum of all positive coefficients is 6+10+10+6 = 32.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Creighton Dement, Floretion Online Multiplier.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,4,-16,24,-16).

Cf. A000079, A009116, A009545, A038503, A077957, A077966, A100212, A100215, A100216.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-4*x+6*x^2-4*x^3)*(1-4*x^4)) )); // G. C. Greubel, Mar 29 2024

Rest[CoefficientList[Series[x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x)(1-2x^2)(1-2x+2x^2)(1+2x^2)),{x,0,40}],x]] (* or *) LinearRecurrence[{4,-6,4,4,-16,24,-16},{4,9,14,18,32,64,128},40] (* Harvey P. Dale, Aug 23 2015 *)

def A100213_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-4*x+6*x^2-4*x^3)*(1-4*x^4)) ).list()
a=A100213_list(51); a[1:] # G. C. Greubel, Mar 29 2024

a(n) = A100215(n) - A100212(n).
a(n) = (-1)^n*A009116(n+3) + A100216 + A038503(n+1).
Equation above in Floretian Algebra operator speak: (pos) + (neg) = (ves) = (jes) + (les) + (tes)
a(n-1) = A000079(n+1) + (5*A077957(n) + 6*A077957(n-1))/4 + A009545(n)/2 + A009545(n+1) + A077966(n-1) - A077966(n)/4. - R. J. Mathar, May 07 2008
From G. C. Greubel, Mar 29 2024: (Start)
a(n) = (1/16)*( 2^(n+4) - 2*((1+5*i)*(1+i)^n + (1-5*i)*(1-i)^n) + (1 - (-1)^n)*2^((n+1)/2)*(5+i^(n+1)) + (1+(-1)^n)*2^(1+n/2)*(3-2*i^n) ).
a(2*n-1) = 2^(n-3)*( 2^(n+2) + 5 + (-1)^n - 6*cos(n*Pi/2) + 4*sin(n*Pi/2) ), for n >= 1.
a(2*n) = 2^(n-2)*( 2^(n+2) + 3 - 2*(-1)^n - cos(n*Pi/2) + 5*sin(n*Pi/2) ), n >= 1.
E.g.f.: -1 + exp(2*x) + (1/8)*(6*cosh(sqrt(2)*x) + 5*sqrt(2)* sinh(sqrt(2)*x) - (4*cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x)) - 2*exp(x)*(cos(x) - 5*sin(x)) ). (End)

Replaced definition with generating function, changed offset to 1. - R. J. Mathar, Mar 12 2010

A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 9, 6, 1, 8, 24, 25, 10, 1, 16, 60, 85, 55, 15, 1, 32, 144, 258, 231, 105, 21, 1, 64, 336, 728, 833, 532, 182, 28, 1, 128, 768, 1952, 2720, 2241, 1092, 294, 36, 1, 256, 1728, 5040, 8280, 8361, 5301, 2058, 450, 45, 1

Offset: 0

Philippe Deléham, Sep 09 2009

Rows sums: A006012; Diagonal sums: A052960.

The sums of each column of A117317 with its subsequent column, treated as a lower triangular matrix with an initial null column attached, or, equivalently, the products of the row polynomials p(n,y) of A117317 with (1+y) with the initial first row below added to the final result. The reversal of A117317 is A056242 with several combinatorial interpretations. - Tom Copeland, Jan 08 2017

			Triangle begins:
  1;
  1,  1;
  2,  3,  1;
  4,  9,  6,  1;
  8, 24, 25, 10,  1; ...

Cf. A000012, A000217, A005582, A011782, A084858.

Cf. A056242, A117317.

Sum_{k=0..n} T(n,k)*x^k = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. Sum_{k=0..n} T(n,k)*x^(n-k) = A123335(n), A000007(n), A000012(n), A006012(n), A084120(n), A090965(n), A165225(n), A165229(n), A165230(n), A165231(n), A165232(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively.
G.f.: (1-(1+y)*x)/(1-2(1+y)*x+(y+y^2)*x^2). - Philippe Deléham, Dec 19 2011
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if nPhilippe Deléham, Dec 19 2011

O.g.f. corrected by Tom Copeland, Jan 15 2017

A009014 Expansion of E.g.f.: (1 + x)/(1 + x + x^2/2).

Original entry on oeis.org

1, 0, -1, 3, -6, 0, 90, -630, 2520, 0, -113400, 1247400, -7484400, 0, 681080400, -10216206000, 81729648000, 0, -12504636144000, 237588086736000, -2375880867360000, 0, 548828480360160000, -12623055048283680000

Offset: 0

R. H. Hardin

G. C. Greubel, Table of n, a(n) for n = 0..450
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

Cf. A009116, A009769.

seq(coeff(series(factorial(n)*((1+x)/(1+x+x^2/2)), x,n+1),x,n),n=0..25); # Muniru A Asiru, Jul 21 2018

nn = 26; Range[0, nn]! CoefficientList[Series[1/Cosh[Log[1 + x]], {x, 0, nn}], x] (* T. D. Noe, Oct 05 2011 *)

PARI
```
a(n)=-(2^-n)*n!*real((-1+I)^(n+1))
```

x='x+O('x^30); Vec(serlaplace(1/cosh(log(1+x)))) \\ G. C. Greubel, Jul 21 2018

E.g.f.: (1+x)/(1+x+x^2/2) = 1/cosh(log(1+x)).
a(n) = -n*a(n-1)-n*(n-1)/2*a(n-2), a(0)=1, a(1)=0.
From Peter Bala, Oct 05 2011: (Start)
a(n) = -n!*(((-1+i)/2)^(n+1) + ((-1-i)/2)^(n+1)) = -n!/sqrt(2)^(n-1)*cos(3*Pi*(n+1)/4).
a(n) = 1/2^n*A009769(n) = -n!/2^n*A009116(n+1).
a(n) = 2*(-1)^n*int {t = 0..inf} t^n*exp(-t)*cos(t). (End)

Extended with signs by Olivier Gérard, Mar 15 1997

Better description from Michael Somos

Previous Showing 11-20 of 34 results. Next

cp's OEIS Frontend

A057681 a(n) = Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A104597 Triangle T read by rows: inverse of Motzkin triangle A097609.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Maxima

Formula

A084102 Inverse binomial transform of A084101.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A090132 Expansion of (1+2*x)/(1+2*x+2*x^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

PARI

Formula

A115416 Imaginary part of (n + i)^n, with i=sqrt(-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A318438 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let h(n) = Sum_{k=0..w} b_k * (i-1)^k (where i denotes the imaginary unit); a(n) is the real part of h(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A057681 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j).

A090132 Expansion of (1+2x)/(1+2x+2*x^2).

A100213 Expansion of g.f.: x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x) (1-2x^2) (1-2x+2x^2) * (1+2*x^2)).