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 1-6 of 6 results.

A105810 Inverse of a Fibonacci-Pascal matrix A105809.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 1, 2, -3, 1, -2, -1, 5, -4, 1, 3, -1, -6, 9, -5, 1, -4, 4, 5, -15, 14, -6, 1, 5, -8, -1, 20, -29, 20, -7, 1, -6, 13, -7, -21, 49, -49, 27, -8, 1, 7, -19, 20, 14, -70, 98, -76, 35, -9, 1, -8, 26, -39, 6, 84, -168, 174, -111, 44, -10, 1, 9, -34, 65, -45, -78, 252, -342, 285, -155, 54, -11, 1
Offset: 0

Views

Author

Paul Barry, May 04 2005

Keywords

Comments

First column is A105811, row sums are A105812, antidiagonal sums are (-1)^n.

Examples

			The triangle T(n,m) begins:
n\m   0   1   2    3   4    5    6    7     8    9   10  11  12 13 ...
0:    1
1:   -1   1
2:    0  -2   1
3:    1   2  -3    1
4:   -2  -1   5   -4   1
5:    3  -1  -6    9  -5    1
6:   -4   4   5  -15  14   -6    1
7:    5  -8  -1   20 -29   20   -7    1
8:   -6  13  -7  -21  49  -49   27   -8     1
9:    7 -19  20   14 -70   98  -76   35    -9    1
10:  -8  26 -39    6  84 -168  174 -111    44  -10    1
11:   9 -34  65  -45 -78  252 -342  285  -155   54  -11   1
12: -10  43 -99  110  33 -330  594 -627   440 -209   65 -12   1
13:  11 -53 142 -209  77  363 -924 1221 -1067  649 -274  77 -13  1
... Reformatted and extended - _Wolfdieter Lang_, Oct 04 2014
-----------------------------------------------------------------------
Recurrence for T(n, 0) with row n-1 entries from Z-sequence (see a link given above): 3 = T(5, 0) = -(1*(-2) + 1*(-1) + 1*5 + 2*(-4) + 3*1) = 3.

Crossrefs

Cf. A105809, A105811, A105812, A248155 (alternating row sum). - Wolfdieter Lang, Oct 04 2014

Programs

  • Maple
    C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if
end proc:
for n from 0 to 10 do
    seq((-1)^(n+k)*(C(n, n-k) - add(C(n-i, n-k-i), i = 2..n)), k = 0..n);
end do; # Peter Bala, Mar 21 2018

Formula

Riordan array ((1+x-x^2)/(1+x)^2, x/(1+x)); Number triangle T(n, 0)=A105811(n), T(n, m)=-T(n-1, m-1)+T(n-1, m).
From Wolfdieter Lang, Oct 04 2014: (Start)
O.g.f. for row polynomials R(n,x) = sum(T(n,m)*x^m,m=0..n): (1 + z - z^2)/((1+z)*(1+(1-x)*z)) (Riordan property).
O.g.f. column m: x^m*(1 + x - x^2)/(1 + x)^(m+2), m >= 0.
The A-sequence of this Riordan triangle is [1, -1]. See the above given recurrence for T(n,m) for n>=1. The Z-sequence has o.g.f. -(1 - x^2)/(1 - x - x^2) and is -A132916(n+5) = -[1, 1, 1, 2, 3, 5, 8, 13, 21, 34,...]. See the W. Lang link under A006232 for Riordan A- and Z-sequences. (End)
T(n,k) = (-1)^(n+k)*(C(n, n-k) - Sum_{i = 2..n} C(n-i, n-k-i)), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0. - Peter Bala, Mar 21 2018

A147678 Double, add 0, double, add 1, double, add 2, double, add 3, etc.

Original entry on oeis.org

1, 2, 2, 4, 5, 10, 12, 24, 27, 54, 58, 116, 121, 242, 248, 496, 503, 1006, 1014, 2028, 2037, 4074, 4084, 8168, 8179, 16358, 16370, 32740, 32753, 65506, 65520, 131040, 131055, 262110, 262126, 524252, 524269, 1048538, 1048556, 2097112, 2097131
Offset: 1

Views

Author

N. J. A. Sloane, Apr 21 2009

Keywords

Comments

A147675-A147678 are from a quiz that someone asked me to help them with.

Links

Programs

  • Mathematica
    Transpose[NestList[Flatten[{Rest[#],2First[#]-5#[[3]]+ 4#[[5]]}]&,{1,2,2,4,5,10},40]][[1]] (* Harvey P. Dale, Mar 24 2011 *)
dp[a_,n_]:=Flatten[{{x=a},Table[{2x,x=2x+m},{m,0,n}]}]; A147678=dp[1,20] (* Zak Seidov, Mar 24 2011 *)
LinearRecurrence[{0, 4, 0, -5, 0, 2}, {1, 2, 2, 4, 5, 10}, 20] (* T. D. Noe, Mar 25 2011 *)

Formula

From R. J. Mathar, Apr 22 2009: (Start)
a(n) = 4*a(n-2) - 5*a(n-4) + 2*a(n-6).
G.f.: -x*(2*x+1)*(2*x^4 - 2*x^2 + 1)/((2*x^2-1)*(x-1)^2*(1+x)^2).
a(n)=(1 + (-1)^n)/2 + 2*A016116(n) - A105811(n+3)/4 - 3*(n+1)/4. (End)
a(n) = 2*a(n-1) - (n mod 2)*(a(n-1) - (n-3)/2). - Reinhard Zumkeller, Apr 22 2009

Extensions

More terms from R. J. Mathar, Apr 22 2009

A124791 Row sums of number triangle A124790.

Original entry on oeis.org

1, 1, 1, 3, 5, 13, 29, 73, 181, 465, 1205, 3171, 8425, 22597, 61073, 166195, 454949, 1251985, 3461573, 9611191, 26787377, 74916661, 210178457, 591347989, 1668172841, 4717282753, 13369522249, 37970114703, 108045430901
Offset: 0

Views

Author

Paul Barry, Nov 07 2006

Keywords

Comments

Row sums of a generalized Motzkin triangle.
Apparently the Motzkin transform of A105811, after the sign of A105811(1) is negated. - R. J. Mathar, Dec 11 2008
Hankel transform is A159964. - Paul Barry, Apr 28 2009

Links

Crossrefs

Cf. A105811, A124790, A159964.

Formula

Conjecture: (n+1)*a(n) +(-n+2)*a(n-1) +(-5*n+7)*a(n-2) +3*(-n+2)*a(n-3) = 0. - R. J. Mathar, Dec 02 2014

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

Views

Author

Paul Curtz, Sep 19 2007

Keywords

Comments

Also the main diagonal of A138057.

Links

Crossrefs

Cf. A105811.
Cf. A181983 (main entry).

Programs

Formula

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)

Extensions

Edited by R. J. Mathar, Jul 07 2008

A173247 a(0) = -1 and a(n) = (-1)^n*(n - 4 - 3*n^2)/2 for n >= 1.

Original entry on oeis.org

-1, 3, -7, 14, -24, 37, -53, 72, -94, 119, -147, 178, -212, 249, -289, 332, -378, 427, -479, 534, -592, 653, -717, 784, -854, 927, -1003, 1082, -1164, 1249, -1337, 1428, -1522, 1619, -1719, 1822, -1928, 2037, -2149, 2264, -2382, 2503, -2627, 2754
Offset: 0

Views

Author

Roger L. Bagula, Feb 13 2010

Keywords

Links

Crossrefs

Cf. A089071, A105811, A143689.

Programs

  • Magma
    [-1] cat [(-1)^n*(n-4-3*n^2)/2: n in [1..50]]; // Vincenzo Librandi, Apr 20 2015
  • Mathematica
    p[x_] = (x^3 - x^2 - 1)/(x + 1)^3;
a = Table[SeriesCoefficient[ Series[p[x], {x, 0, 50}], n], {n, 0, 50}]
  • PARI
    Vec((x^3 - x^2 - 1)/(x + 1)^3 + O(x^50)) \\ Michel Marcus, Apr 20 2015

Formula

G.f.: (x^3 - x^2 - 1)/(x + 1)^3.
a(n) = -3*a(n-1) -3*a(n-2) -a(n-3).
From Franck Maminirina Ramaharo, Dec 27 2018: (Start)
a(n) = (A143689(n) + 1)*(-1)^(n + 1), n >= 1.
E.g.f.: 1 - (1/2)*(4 - 2*x + 3*x^2)*exp(-x). (End)

Extensions

Definition simplified by the Assoc. Editors of the OEIS, Feb 21 2010
Incorrect comment removed by Joerg Arndt, Dec 27 2018

A173248 a(0)=1, a(n) = (-1)^n*(n^3-15*n^2-12+2*n)/6, n>0.

Original entry on oeis.org

-1, 4, -10, 19, -30, 42, -54, 65, -74, 80, -82, 79, -70, 54, -30, -3, 46, -100, 166, -245, 338, -446, 570, -711, 870, -1048, 1246, -1465, 1706, -1970, 2258, -2571, 2910, -3276, 3670, -4093, 4546, -5030, 5546, -6095, 6678, -7296, 7950, -8641, 9370
Offset: 0

Views

Author

Roger L. Bagula, Feb 13 2010

Keywords

Comments

Limiting ratio a(n+1)/a(n) is near -1.071806167400881 as n->infinity.

Links

Crossrefs

Cf. A089071, A105811

Programs

  • Mathematica
    p[x_] = (x^4 - x^3 - 1)/(x + 1)^4;
a = Table[SeriesCoefficient[ Series[p[x], {x, 0, 50}], n], {n, 0, 50}]
LinearRecurrence[{-4,-6,-4,-1},{-1,4,-10,19,-30},50] (* Harvey P. Dale, Nov 21 2019 *)

Formula

G.f.: (x^4 - x^3 - 1)/(x + 1)^4.
a(n)= -4*a(n-1) -6*a(n-2) -4*a(n-3) -a(n-4).

Extensions

Definition simplified by the Assoc. Editors of the OEIS, Feb 21 2010
Showing 1-6 of 6 results.

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

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