cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A091699 Row sums of triangle A091698.

Original entry on oeis.org

1, 0, -1, 3, -9, 30, -109, 420, -1685, 6960, -29391, 126291, -550359, 2426502, -10803801, 48507843, -219377949, 998436792, -4569488371, 21016589073, -97090411019, 450314942682, -2096122733211, 9788916220518
Offset: 0

Views

Author

Christian G. Bower, Jan 29 2004

Keywords

Comments

Hankel transform is (-1)^n. - Paul Barry, Jun 17 2010

Links

Crossrefs

Cf. A091698.

Programs

  • Mathematica
    CoefficientList[Series[2/(3+3*x-Sqrt[1+6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
  • PARI
    a(n)=polcoeff(2/(3+3*x-sqrt(1+6*x+5*x^2+x*O(x^n))),n,x) \\ Paul D. Hanna

Formula

G.f.: A(x) = 2/(3+3*x - sqrt(1+6*x+5*x^2)). - Paul D. Hanna, Feb 23 2004
Conjecture: 2*n*a(n) + (13*n-16)*a(n-1) + 4*(4*n-7)*a(n-2) + 5*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ (-1)^(n+1) * 5^(n+3/2) / (72 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014

A063967 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 5, 15, 16, 7, 1, 8, 30, 43, 29, 9, 1, 13, 58, 104, 95, 46, 11, 1, 21, 109, 235, 271, 179, 67, 13, 1, 34, 201, 506, 705, 591, 303, 92, 15, 1, 55, 365, 1051, 1717, 1746, 1140, 475, 121, 17, 1, 89, 655, 2123, 3979, 4759, 3780, 2010, 703, 154, 19, 1
Offset: 0

Views

Author

Henry Bottomley, Sep 05 2001

Keywords

Examples

			T(3,1) = T(2,1) + T(1,1) + T(2,0) + T(1,0) = 3 + 1 + 2 + 1 = 7.
Triangle begins:
   1,
   1,   1,
   2,   3,   1,
   3,   7,   5,   1,
   5,  15,  16,   7,   1,
   8,  30,  43,  29,   9,   1,
  13,  58, 104,  95,  46,  11,  1,
  21, 109, 235, 271, 179,  67, 13,  1,
  34, 201, 506, 705, 591, 303, 92, 15, 1

Links

Crossrefs

Row sums are A002605.
Columns include: A000045(n+1), A023610(n-1).
Main diagonal: A000012, a(n, n-1) = A005408(n-1).
Matrix inverse: A091698, matrix square: A091700.
Cf. A321620.
Sum_{k=0..n} x^k*T(n,k) is (-1)^n*A057086(n) (x=-11), (-1)^n*A057085(n+1) (x=-10), (-1)^n*A057084(n) (x=-9), (-1)^n*A030240(n) (x=-8), (-1)^n*A030192(n) (x=-7), (-1)^n*A030191(n) (x=-6), (-1)^n*A001787(n+1) (x=-5), A000748(n) (x=-4), A108520(n) (x=-3), A049347(n) (x=-2), A000007(n) (x=-1), A000045(n) (x=0), A002605(n) (x=1), A030195(n+1) (x=2), A057087(n) (x=3), A057088(n) (x=4), A057089(n) (x=5), A057090(n) (x=6), A057091(n) (x=7), A057092(n) (x=8), A057093(n) (x=9). - Philippe Deléham, Nov 03 2006

Programs

  • Haskell
    a063967_tabl = [1] : [1,1] : f [1] [1,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) ([0] ++ us ++ [0]) $
          zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 17 2013
  • Mathematica
    T[n_, k_] := Sum[Binomial[j, n - j]*Binomial[j, k], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 11 2017, after Paul Barry *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1/(1 - x - x^2), 11] // Flatten (* Peter Luschny, Nov 27 2018 *)

Formula

G.f.: 1/(1-x*(1+x)*(1+y)). - Vladeta Jovovic, Oct 11 2003
Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)). The inverse of the signed version (1/(1+x-x^2),x(1-x)/(1+x-x^2)) is abs(A091698). - Paul Barry, Jun 10 2005
T(n, k) = Sum_{j=0..n} C(j, n-j)C(j, k). - Paul Barry, Nov 09 2005
Diagonal sums are A002478. - Paul Barry, Nov 09 2005
A026729*A007318 as infinite lower triangular matrices. - Philippe Deléham, Dec 11 2008
Central coefficients T(2*n,n) are A137644. - Paul Barry, Apr 15 2010
Product of Riordan arrays (1, x(1+x))*(1/(1-x), x/(1-x)), that is, A026729*A007318. - Paul Barry, Mar 14 2011
Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 12 2011

A050511 a(n) = (-1)^n * Sum_{i=0..n} binomial(n+1,i+1)*Catalan(i).

Original entry on oeis.org

1, -3, 8, -23, 74, -262, 993, -3943, 16178, -68000, 291191, -1265618, 5568263, -24749363, 110961248, -501209303, 2278704938, -10419244888, 47882934663, -221047167628, 1024586641973, -4766517165713, 22248226873538, -104160733650738, 489007907489239
Offset: 0

Views

Author

N. J. A. Sloane, Dec 28 1999

Keywords

References

  • J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 371.

Links

Crossrefs

Column 1 of A091698.

Programs

  • Mathematica
    Table[(-1)^n Sum[Binomial[n+1,i+1]CatalanNumber[i],{i,0,n}],{n,0,30}] (* Harvey P. Dale, Jan 26 2013 *)

Formula

G.f.: (sqrt((5*x+1)/(x+1))-1)/(2*x*(x+1)).
(n+1)*a(n)=(1-6*n)*a(n-1)-5*n*a(n-2)+(-1)^n. - Vladeta Jovovic, Jul 16 2004
a(n) ~ (-1)^n * 5^(n+5/2) / (32 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 02 2014

A168216 Riordan array (1/(1-x),xc(x)/(1-xc(x))) where c(x)is the g.f. of A000108.It factorizes as A007318*A106566.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 23, 19, 7, 1, 1, 74, 69, 34, 9, 1, 1, 262, 256, 147, 53, 11, 1, 1, 993, 986, 615, 265, 76, 13, 1, 1, 3943, 3935, 2571, 1235, 431, 103, 15, 1, 1, 16178, 16169, 10862, 5591, 2216, 653, 134, 17, 1
Offset: 0

Views

Author

Philippe Deléham, Nov 20 2009

Keywords

Comments

Inverse is Riordan array (1/(1+x-x^2),x(1-x)/(1+x-x^2)) = [ -1,-1,1,0,0,0,0,0,...]DELTA[1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A091698.

Examples

			Triangle begins : 1 ; 1,1 ; 1,3,1 ; 1,8,5,1 ; 1,23,19,7,1 ; ...

Crossrefs

Cf. A000108, A007318, A106566.

Formula

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000012(n), A007317(n+1), A026375(n) for x = 0, 1, 2 respectively.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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