A057077 -id:A057077 - cp's OEIS frontend

A195851 Column 7 of array A195825. Also column 1 of triangle A195841. Also 1 together with the row sums of triangle A195841.

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 87, 89, 95, 107, 128, 152, 173, 185, 192, 196, 203, 216, 242, 281, 328, 367, 394, 409, 421, 436, 465

Offset: 0

Omar E. Pol, Oct 07 2011

nonn

Note that this sequence contains four plateaus: [1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4], [13, 13, 13, 13], [35, 35]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Cf. A000041, A001082, A006950, A036820, A057077, A195160, A195831, A195825, A195848, A195849, A195850, A195852, A196933, A210964, A211971.

A195160 := proc(n)
        (18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16 ;
end proc:
A195841 := proc(n, k)
        option remember;
        local ks, a, j ;
        if A195160(k) > n then
                0 ;
        elif n <= 5 then
                return 1;
        elif k = 1 then
                a := 0 ;
                for j from 1 do
                        if A195160(j) <= n-1 then
                                a := a+procname(n-1, j) ;
                        else
                                break;
                        end if;
                end do;
                return a;
        else
                ks := A195160(k) ;
                (-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
        end if;
end proc:
A195851 := proc(n)
        A195841(n+1,1) ;
end proc:
seq(A195851(n), n=0..60) ; # R. J. Mathar, Oct 08 2011

G.f.: Product_{k>=1} 1/((1 - x^(9*k))*(1 - x^(9*k-1))*(1 - x^(9*k-8))). - Ilya Gutkovskiy, Aug 13 2017

a(n) ~ exp(Pi*sqrt(2*n)/3) / (8*sin(Pi/9)*n). - Vaclav Kotesovec, Aug 14 2017

A196933 Column 9 of array A195825. Also column 1 of triangle A195843. Also 1 together with the row sums of triangle A195843.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 195, 197, 203, 216, 242, 281

Offset: 0

Omar E. Pol, Oct 07 2011

nonn

Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13], [35, 35, 35, 35], [86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

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

Cf. A000041, A001082, A006950, A036820, A057077, A195313, A195825, A195833, A195848, A195849, A195850, A195851, A195852, A210964, A211971.

T := Product[1/((1 - x^(11*k))*(1 - x^(11*k - 1))*(1 - x^(11*k - 10))), {k, 1, 70}]; a:= CoefficientList[Series[T, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 28 2018 *)

G.f.: Product_{k>=1} 1/((1 - x^(11*k))*(1 - x^(11*k-1))*(1 - x^(11*k-10))). - Ilya Gutkovskiy, Aug 13 2017

a(n) ~ exp(Pi*sqrt(2*n/11)) / (8*sin(Pi/11)*n). - Vaclav Kotesovec, Aug 14 2017

More terms from Omar E. Pol, Jun 10 2012

A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Jun 10 2012

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A211970
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.        (if k>=1)                         with a(0)=1,
.                                          if k >= 0]
========================================================
0    4    A008794        -         -         A211971
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
2,     2,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
4,     3,   2,   1,   1,   1,  1,  1,  1,  1,  1, ...
6,     5,   3,   2,   1,   1,  1,  1,  1,  1,  1, ...
10,    7,   4,   3,   2,   1,  1,  1,  1,  1,  1, ...
16,   11,   5,   4,   3,   2,  1,  1,  1,  1,  1, ...
24,   15,   7,   4,   4,   3,  2,  1,  1,  1,  1, ...
36,   22,  10,   5,   4,   4,  3,  2,  1,  1,  1, ...
54,   30,  13,   7,   4,   4,  4,  3,  2,  1,  1, ...
78,   42,  16,  10,   5,   4,  4,  4,  3,  2,  1, ...
112,  56,  21,  12,   7,   4,  4,  4,  4,  3,  2, ...
160,  77,  28,  14,  10,   5,  4,  4,  4,  4,  3, ...
224, 101,  35,  16,  12,   7,  4,  4,  4,  4,  4, ...
312, 135,  43,  21,  13,  10,  5,  4,  4,  4,  4, ...
432, 176,  55,  27,  14,  12,  7,  4,  4,  4,  4, ...
...

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A195825.
Cf. A057077, A195152.

T(n,k) = A211971(n), if k = 0.

T(n,k) = A195825(n,k), if k >= 1.

A002621 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)*(1-x^4)).

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 27, 38, 53, 71, 94, 121, 155, 194, 241, 295, 359, 431, 515, 609, 717, 837, 973, 1123, 1292, 1477, 1683, 1908, 2157, 2427, 2724, 3045, 3396, 3774, 4185, 4626, 5104, 5615, 6166, 6754, 7386, 8058, 8778, 9542, 10358, 11222, 12142, 13114

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 199
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1).

Partial sums of A001400. Column 4 of A092905.

A002621:=-1/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**5; # Simon Plouffe in his 1992 dissertation
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+2), right=Set(U, card=1)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=4..51) ; # Zerinvary Lajos, Feb 07 2008
A057077 := proc(n) (-1)^floor(n/2) ; end proc:
A061347 := proc(n) op(1+(n mod 3),[1,1,-2]) ; end proc:
A002621 := proc(n) 83/288*n^2+55/64*n+2815/3456+11/288*n^3+1/576*n^4+11/128*(-1)^n+1/64*(-1)^n*n; %+ A057077(n)/16 +A061347(n)/27; end proc:
seq(A002621(n),n=0..10) ; # R. J. Mathar, Mar 15 2011

CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^4)),{x,0,60}],x] (* Stefan Steinerberger, Jun 10 2007 *)
LinearRecurrence[{2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1}, {1, 2, 4, 7, 12, 18, 27, 38, 53, 71, 94}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

a(n)=(n+1)*(9*(-1)^n+n^3+21*n^2+145*n+350)\/576 \\ Charles R Greathouse IV, May 23 2013

a(n) = +2*a(n-1) - a(n-3) - 2*a(n-5) + 2*a(n-6) + a(n-8) - 2*a(n-10) + a(n-11).
a(n) = 83*n^2/288 +55*n/64 +2815/3456 +11*n^3/288 +n^4/576 +11*(-1)^n/128 +(-1)^n*n/64 + A057077(n)/16 +A061347(n)/27. - R. J. Mathar, Mar 15 2011
a(n)=floor((n+1)*(9*(-1)^n + n^3 + 21*n^2 + 145*n + 350)/576 + 1/2). - Tani Akinari, Nov 10 2012

A143623 Decimal expansion of the constant cos(1) + sin(1).

Original entry on oeis.org

1, 3, 8, 1, 7, 7, 3, 2, 9, 0, 6, 7, 6, 0, 3, 6, 2, 2, 4, 0, 5, 3, 4, 3, 8, 9, 2, 9, 0, 7, 3, 2, 7, 5, 6, 0, 3, 3, 5, 4, 8, 7, 3, 4, 8, 1, 4, 1, 6, 2, 9, 3, 2, 9, 3, 3, 4, 2, 8, 4, 8, 9, 6, 5, 3, 7, 3, 0, 1, 0, 7, 9, 9, 1, 6, 5, 7, 1, 1, 4, 3, 3, 4, 6, 6, 5, 9, 1, 5, 9, 9, 6, 3, 0, 2, 3, 5, 7, 8, 5, 1

Offset: 1

Peter Bala, Aug 30 2008

cos(1) + sin(1) = Sum_{n >= 0} (-1)^floor(n/2)/n! = 1 + 1/1! - 1/2! - 1/3! + 1/4! + 1/5! - 1/6! - 1/7! + + - - ... .
Define E_2(k) = Sum_{n >= 0} (-1)^floor(n/2)*n^k/n! for k = 0, 1, 2, ... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1).
Furthermore, E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = E_2(1) - E_2(0), E_2(3) = -3*E_2(0) and E_2(4) = -5*E_2(1) - 6*E_2(0). The precise result is E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1).
The decimal expansion of the constant cos(1) - sin(1) = E_2(1) is recorded in A143624. Compare with A143625.

			1.38177329067603622405 ... .

Cf. A049469, A049470, A057077, A121867, A121868, A143624, A143625.

RealDigits[Cos[1]+Sin[1],10,120][[1]] (* Harvey P. Dale, Mar 01 2019 *)

sqrt(2)*sin(1+Pi/4) \\ Charles R Greathouse IV, Feb 03 2025

Equals sin(1+Pi/4)*sqrt(2). - Franklin T. Adams-Watters, Jun 27 2014

Offset corrected by R. J. Mathar, Feb 05 2009

A195852 Column 8 of array A195825. Also column 1 of triangle A195842. Also 1 together with the row sums of triangle A195842.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 194, 197, 203, 216, 242, 281, 328, 367, 393, 407

Offset: 0

Omar E. Pol, Oct 07 2011

Note that this sequence contains four plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13], [35, 35, 35]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Number of partitions of n into parts congruent to 0, 1 or 9 (mod 10). - Peter Bala, Dec 10 2020

Cf. A000041, A001082, A006950, A036820, A057077, A195162, A195825, A195832, A195848, A195849, A195850, A195851, A196933, A210964, A211971.

G.f.: Product_{k>=1} 1/((1 - x^(10*k))*(1 - x^(10*k-1))*(1 - x^(10*k-9))). - Ilya Gutkovskiy, Aug 13 2017
a(n) ~ exp(Pi*sqrt(n/5))/(2*(sqrt(5)-1)*n). - Vaclav Kotesovec, Aug 14 2017
a(n) = a(n-1) + a(n-9) - a(n-12) - a(n-28) + + - - (with the convention a(n) = 0 for negative n), where 1, 9, 12, 28, ... is the sequence of generalized 12-gonal numbers A195162. - Peter Bala, Dec 10 2020

More terms from Omar E. Pol, Jun 10 2012

A129818 Riordan array (1/(1+x), x/(1+x)^2), inverse array is A039599.

Original entry on oeis.org

1, -1, 1, 1, -3, 1, -1, 6, -5, 1, 1, -10, 15, -7, 1, -1, 15, -35, 28, -9, 1, 1, -21, 70, -84, 45, -11, 1, -1, 28, -126, 210, -165, 66, -13, 1, 1, -36, 210, -462, 495, -286, 91, -15, 1, -1, 45, -330, 924, -1287, 1001, -455, 120, -17, 1, 1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1

Offset: 0

Philippe Deléham, Jun 09 2007

This sequence is up to sign the same as A129818. - T. D. Noe, Sep 30 2011
Row sums: A057078. - Philippe Deléham, Jun 11 2007
Subtriangle of the triangle given by (0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012
This triangle provides the coefficients of powers of x^2 for the even-indexed Chebyshev S polynomials (see A049310): S(2*n,x) = Sum_{k=0..n} T(n,k)*x^(2*k), n >= 0. - Wolfdieter Lang, Dec 17 2012
If L(x^n) := C(n) = A000108(n) (Catalan numbers), then the polynomials P_n(x) := Sum_{k=0..n} T(n,k)*x^k are orthogonal with respect to the inner product given by (f(x),g(x)) := L(f(x)*g(x)). - Michael Somos, Jan 03 2019

			Triangle T(n,k) begins:
  n\k  0   1    2     3     4     5    6    7    8   9 10 ...
   0:  1
   1: -1   1
   2:  1  -3    1
   3: -1   6   -5     1
   4:  1 -10   15    -7     1
   5: -1  15  -35    28    -9     1
   6:  1 -21   70   -84    45   -11    1
   7: -1  28 -126   210  -165    66  -13    1
   8:  1 -36  210  -462   495  -286   91  -15    1
   9: -1  45 -330   924 -1287  1001 -455  120  -17   1
  10:  1 -55  495 -1716  3003 -3003 1820 -680  153 -19  1
  ... Reformatted by _Wolfdieter Lang_, Dec 17 2012
Recurrence from the A-sequence A115141:
15 = T(4,2) = 1*6 + (-2)*(-5) + (-1)*1.
(0, -1, 0, -1, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, ...) begins:
  1
  0,  1
  0, -1,   1
  0,  1,  -3,   1
  0, -1,   6,  -5,  1
  0,  1, -10,  15, -7,  1
  0, -1,  15, -35, 28, -9, 1. - _Philippe Deléham_, Mar 19 2012
Row polynomial for n=3 in terms of x^2: S(6,x) = -1 + 6*x^2 -5*x^4 + 1*x^6, with Chebyshev's S polynomial. See a comment above. - _Wolfdieter Lang_, Dec 17 2012
Boas-Buck type recurrence: -35 = T(5,2) = (5/3)*(-1*1 +1*(-5) - 1*15) = -3*7 = -35. - _Wolfdieter Lang_, Jun 03 2020

Vincenzo Librandi, Rows n = 1..101, flattened
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry and Aoife Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, example 15.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Cf. A039599, A085478, A123970, A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare((1 - sqrt(1 - 4*x))/(2*x), 10):
LinearAlgebra[MatrixInverse](%); # Peter Luschny, Jan 04 2019

max = 10; Flatten[ CoefficientList[#, y] & /@ CoefficientList[ Series[ (1 + x)/(1 + (2 - y)*x + x^2), {x, 0, max}], x]] (* Jean-François Alcover, Sep 29 2011, after Wolfdieter Lang *)

@CachedFunction
def A129818(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = A129818(n-1,k) if n==1 else 2*A129818(n-1,k)
    return A129818(n-1,k-1) - A129818(n-2,k) - h
for n in (0..9): [A129818(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

T(n,k) = (-1)^(n-k)*A085478(n,k) = (-1)^(n-k)*binomial(n+k,2*k).
Sum_{k=0..n} T(n,k)*A000531(k) = n^2, with A000531(0)=0. - Philippe Deléham, Jun 11 2007
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A057078(n), A057077(n), A057079(n), A005408(n), A002878(n), A001834(n), A030221(n), A002315(n), A033890(n), A057080(n), A057081(n), A054320(n), A097783(n), A077416(n), A126866(n), A028230(n+1) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16, respectively. - Philippe Deléham, Nov 19 2009
O.g.f.: (1+x)/(1+(2-y)*x+x^2). - Wolfdieter Lang, Dec 15 2010
O.g.f. column k with leading zeros (Riordan array, see NAME): (1/(1+x))*(x/(1+x)^2)^k, k >= 0. - Wolfdieter Lang, Dec 15 2010
From Wolfdieter Lang, Dec 20 2010: (Start)
Recurrences from the Z- and A-sequences for Riordan arrays. See the W. Lang link under A006232 for details and references.
T(n,0) = -1*T(n-1,0), n >= 1, from the o.g.f. -1 for the Z-sequence (trivial result).
T(n,k) = Sum_{j=0..n-k} A(j)*T(n-1,k-1+j), n >= k >= 1, with A(j):= A115141(j) = [1,-2,-1,-2,-5,-14,...], j >= 0 (o.g.f. 1/c(x)^2 with the A000108 (Catalan) o.g.f. c(x)). (End)
T(n,k) = (-1)^n*A123970(n,k). - Philippe Deléham, Feb 18 2012
T(n,k) = -2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = -1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 19 2012
A039599(m,n) = Sum_{k=0..n} T(n,k) * C(k+m) where C(n) are the Catalan numbers. - Michael Somos, Jan 03 2019
Equals the matrix inverse of the Riordan square (cf. A321620) of the Catalan numbers. - Peter Luschny, Jan 04 2019
Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment in A046521 with references): T(n,k) = ((1 + 2*k)/(n - k))*Sum_{j = k..n-1} (-1)^(n-j)*T(j,k), with input T(n,n) = 1, and T(n,k) = 0 for n < k. - Wolfdieter Lang, Jun 03 2020

A176742 Expansion of (1 - x^2) / (1 + x^2) in powers of x.

Original entry on oeis.org

1, 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, 2

Offset: 0

Wolfdieter Lang, Oct 15 2010

Difference sequence of A057077.
Sequence of determinants of matrices for some bipartite graphs, called Tz(n). The graph Tz(4) appears in the logo for the beer called Tannenzäpfle (small fir cone), brewed by Badische Staatsbrauerei Rothaus, Germany, hence the name Tz. See the link for this logo with Tz(4).
The vertex-vertex matrix for these bipartite graphs will also be called Tz(n) (without leading to confusion).
General proof by expanding the determinant a(n) = determinant(Tz(n)) along the first column yielding b(n-1)-b(n-2), with b(n-1) the A_{1,1} minor of the matrix Tz(n), and deriving a recurrence for the b(n), namely b(n) = -b(n-2) with inputs b(0) = 1 = b(1). This gives b(n) = A057077(n), n>=0.

			G.f. = 1 - 2*x^2 + 2*x^4 - 2*x^6 + 2*x^8 - 2*x^10 + 2*x^12 - 2*x^14 + 2*x^16 + ...
The bipartite graphs Tz(n) (n>=1) look like |, |X|, |XX|, |XXX|, ... For n>=2 the lines have to be connected to give the 2*n nodes and 2*n edges. The n=1 graph Tz(1) has 2*1=2 nodes and only one edge.
n=1: determinant((1))=1; n=2: determinant(Matrix([[1,1],[1,1]]))=0; n=3: determinant(Matrix([[1,1,0],[1,0,1],[0,1,1]]))=-2; n=4: determinant(Tz(4))=0; etc.

Wolfdieter Lang, Some Tz(n) graphs and matrices.
Rothaus Tannenzäpfle, Tz(4) graph on the Tannenzäpfle logo
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A084099, A057077.

a := n -> 2^signum(n)*(-1)^iquo(n+1,2)*modp(n+1,2);
seq(a(n), n=0..100); # Peter Luschny, Jun 22 2014

Join[{1}, Table[{0, -2, 0, 2}, {26}]] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
a[ n_] := - Boole[n == 0] + {0, -2, 0, 2}[[Mod[ n, 4, 1]]]; (* Michael Somos, May 05 2015 *)
PadRight[{1},120,{2,0,-2,0}] (* Harvey P. Dale, Apr 13 2019 *)

{a(n) = - (n == 0) + [2, 0, -2, 0][n%4 + 1]}; /* Michael Somos, Mar 21 2011 */

def A176742(n): return (2,0,-2,0)[n&3] if n else 1 # Chai Wah Wu, Apr 22 2025

Euler transform of length 4 sequence [0, -2, 0, 1]. - Michael Somos, Mar 21 2011
Moebius transform is length 4 sequence [0, -2, 0, 4]. - Michael Somos, Mar 22 2011
a(n) = a(-n) for all n in Z. a(n) = c_4(n) if n>1, where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
a(n-1) := determinant(Tz(n)), n>=1. The rows of the matrix Tz(4) are [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 0, 1], [0, 0, 1, 1]]. Tz(1)=(1), and Tz(2) has rows [[1, 1], [1, 1]]. The matrix for the generalization Tz(n) has rows [[1,1,0,...,0], [1,0,1,0,...,0], [0,1,0,1,0,...,0], ..., [0,...,0,1,0,1], [0,...,0,1,1].
a(0)=1, a(2*k-1)= 0, a(4*k) = +2, a(4*k-2) = -2, k>=1.
O.g.f.: (1-x^2)/(1+x^2).
a(n) = A057077(n) - A057077(n-1), n>=1. a(0)=1.
Dirichlet g.f. sum_{n>=1} a(n)/n^s = zeta(s)*(4^(1-s)-2^(1-s)). - R. J. Mathar, Apr 11 2011
a(n) = (((n+1) mod 2)+((n+2+sign(n)) mod 2))*(-1)^ceiling(n/2). - Wesley Ivan Hurt, Jun 20 2014

A067994 Hermite numbers.

Original entry on oeis.org

1, 0, -2, 0, 12, 0, -120, 0, 1680, 0, -30240, 0, 665280, 0, -17297280, 0, 518918400, 0, -17643225600, 0, 670442572800, 0, -28158588057600, 0, 1295295050649600, 0, -64764752532480000, 0, 3497296636753920000, 0, -202843204931727360000, 0

Offset: 0

Eric W. Weisstein, Feb 07 2002

sign

|a(n)| is the number of sets of ordered pairs of n labeled elements. - Steven Finch, Nov 14 2021
|a(n)| is the number of square roots of any permutation in S_{2n} whose disjoint cycle decomposition consists of n transpositions, n > 0. For n=2, permutation (1,2)(3,4) in S_4 has exactly |a(2)|=2 square roots: (1,3,2,4) and (1,4,2,3). - Luis Manuel Rivera Martínez, Feb 25 2015
Self-convolution gives A076729(n)*(-1)^n interleaved with zeros. - Vladimir Reshetnikov, Oct 11 2016
Named after the French mathematician Charles Hermite (1822-1901). - Amiram Eldar, Jun 06 2021

			From _Steven Finch_, Nov 14 2021: (Start)
|a(4)| = 12 because the sets of ordered pairs for n = 4 are
  {(1,2),(3,4)}, {(2,1),(3,4)}, {(1,2),(4,3)}, {(2,1),(4,3)},
  {(1,3),(2,4)}, {(3,1),(2,4)}, {(1,3),(4,2)}, {(3,1),(4,2)},
  {(1,4),(3,2)}, {(4,1),(3,2)}, {(1,4),(2,3)}, {(4,1),(2,3)}. (End)

G. C. Greubel, Table of n, a(n) for n = 0..730
Giuseppe Dattoli, Subuhi Khan, and Ujair Ahmad, Hermite numbers and new families of polynomials, arXiv:2503.14930 [math.NT], 2025.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Eric Weisstein's World of Mathematics, Hermite Number.
Wikipedia, Hermite number.

Cf. A097388 (same sequence without zeros).
Cf. A001813, A076729, A126869, A081123.
Cf. A101109 (ordered triples instead of ordered pairs).

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

A067994 := n -> pochhammer(-n, n/2):
seq(A067994(n), n = 0..31); # Peter Luschny, Nov 14 2021

HermiteH[Range[0,50], 0]
With[{nmax=50}, CoefficientList[Series[Exp[-x^2], {x,0,nmax}],x]*Range[0, nmax]!] (* G. C. Greubel, Jun 09 2018 *)

a(n) = polhermite(n, 0); \\ Michel Marcus, Feb 27 2015

x='x+O('x^30); Vec(serlaplace(exp(-x^2))) \\ G. C. Greubel, Jun 09 2018

E.g.f.: exp(-x^2). - Vladeta Jovovic, Aug 24 2002
a(n) = (-1)^(n/2)*n!/(n/2)! if n is even, 0 otherwise. - Mitch Harris, Feb 01 2006
a(n) = -(2*n-2)*a(n-2). - Alexander Karpov, Jul 24 2017
E.g.f.: U(0) where U(k) = 1 - x^2/((2*k+1) - x^2*(2*k+1)/(x^2 - 2*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2012
G.f.: 1/G(0) where G(k) = 1 + 2*x^2*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2012
E.g.f.: E(0)/(1+x) where E(k) = 1 + x/(1 - x/(x - (k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
E.g.f.: E(0)-1, where E(k) = 2 - x^2/(2*k+1 + x^2/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013
a(2*k) = A097388(k), a(2*k+1) = 0. - Joerg Arndt, Oct 12 2016
From Peter Luschny, Nov 14 2021: (Start)
a(n) = A057077(n)*A126869(n)*A081123(n). In particular, a(n) is divisible by floor(n/2)!.
a(n) = Pochhammer(-n, n/2). (End)

A133607 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = -1, with 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, -2, 1, 0, 1, -1, -3, 2, 1, 0, 1, -1, -4, 3, 3, -1, 0, 1, -1, -5, 4, 6, -3, -1, 0, 1, -1, -6, 5, 10, -6, -4, 1, 0, 1, -1, -7, 6, 15, -10, -10, 4, 1, 0, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1, 0, 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1

Offset: 0

Philippe Deléham, Dec 27 2007

Previous name: Triangle T(n,k), 0<=k<=n, read by rows given by [0, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Triangle begins:
  1;
  0, 1;
  0, 1, -1;
  0, 1, -1, -1;
  0, 1, -1, -2, 1;
  0, 1, -1, -3, 2, 1;
  0, 1, -1, -4, 3, 3, -1;
  0, 1, -1, -5, 4, 6, -3, -1;
  0, 1, -1, -6, 5, 10, -6, -4, 1;
  0, 1, -1, -7, 6, 15, -10, -10, 4, 1;
  0, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1;
  0, 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1;
  0, 1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1;
  ...
Triangle A103631 begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 1, 2, 1;
  0, 1, 1, 3, 2, 1;
  0, 1, 1, 4, 3, 3, 1;
  0, 1, 1, 5, 4, 6, 3, 1;
  0, 1, 1, 6, 5, 10, 6, 4, 1;
  0, 1, 1, 7, 6, 15, 10, 10, 4, 1;
  0, 1, 1, 8, 7, 21, 15, 20, 10, 5, 1;
  0, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1;
  0, 1, 1, 10, 9, 36, 28, 56, 35, 35, 15, 6, 1;
  ...
Triangle A108299 begins:
  1;
  1, -1;
  1, -1, -1;
  1, -1, -2, 1;
  1, -1, -3, 2, 1;
  1, -1, -4, 3, 3, -1;
  1, -1, -5, 4, 6, -3, -1;
  1, -1, -6, 5, 10, -6, -4, 1;
  1, -1, -7, 6, 15, -10, -10, 4, 1;
  1, -1, -8, 7, 21, -15, -20, 10, 5, -1;
  1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1;
  1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1;
  ...

Another version is A108299.

Unsigned version is A103631 (T(n,k) = A103631(n,k)*A057077(k)).

m = 13
(* DELTA is defined in A084938 *)
DELTA[Join[{0, 1}, Table[0, {m}]], Join[{1, -2, 1}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)
qStirling2[n_, k_, q_] /; 1 <= k <= n := q^(k-1) qStirling2[n-1, k-1, q] + Sum[q^j, {j, 0, k-1}] qStirling2[n-1, k, q];
qStirling2[n_, 0, _] := KroneckerDelta[n, 0];
qStirling2[0, k_, _] := KroneckerDelta[0, k];
qStirling2[, , _] = 0;
Table[qStirling2[n, k, -1], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 10 2020 *)

from sage.combinat.q_analogues import q_stirling_number2
for n in (0..9):
    print([q_stirling_number2(n,k).substitute(q=-1) for k in [0..n]])
# Peter Luschny, Mar 09 2020

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A057077(n), A010892(n), A000012(n), A001519(n), A001835(n), A004253(n), A001653(n), A049685(n-1), A070997(n-1), A070998(n-1), A072256(n), A078922(n), A077417(n-1), A085260(n), A001570(n-1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 respectively .
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A010892(n), A133631(n), A133665(n), A133666(n), A133667(n), A133668(n), A133669(n), A133671(n), A133672(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively .
G.f.: (1-x+y*x)/(1-x+y^2*x^2). - Philippe Deléham, Mar 14 2012
T(n,k) = T(n-1,k) - T(n-2,k-2), T(0,0) = T(1,1) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(2,2) = -1 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 14 2012

New name from Peter Luschny, Mar 09 2020

cp's OEIS Frontend

A195851 Column 7 of array A195825. Also column 1 of triangle A195841. Also 1 together with the row sums of triangle A195841.

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A196933 Column 9 of array A195825. Also column 1 of triangle A195843. Also 1 together with the row sums of triangle A195843.

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A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

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A002621 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*(1-x^4)).

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A143623 Decimal expansion of the constant cos(1) + sin(1).

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A195852 Column 8 of array A195825. Also column 1 of triangle A195842. Also 1 together with the row sums of triangle A195842.

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A129818 Riordan array (1/(1+x), x/(1+x)^2), inverse array is A039599.

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A176742 Expansion of (1 - x^2) / (1 + x^2) in powers of x.

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A002621 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)*(1-x^4)).