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.

A110506 Riordan array (1/(1-xc(2x)),xc(2x)/(1-xc(2x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 13, 19, 7, 1, 67, 102, 44, 10, 1, 381, 593, 278, 78, 13, 1, 2307, 3640, 1795, 568, 121, 16, 1, 14589, 23231, 11849, 4051, 999, 173, 19, 1, 95235, 152650, 79750, 28770, 7820, 1598, 234, 22, 1, 636925, 1025965, 545680, 204760, 59650, 13642, 2392, 304, 25, 1
Offset: 0

Views

Author

Paul Barry, Jul 24 2005

Keywords

Comments

Deleham triangle Delta(0^n,2-0^n) [see construction in A084938]. The binomial transform of the inverse of this triangle has general element (-2)^(n-k)*C(k,n-k), that is, it is the Riordan array (1,x(1-2x)) [A110509]. Row sums are A052701. Diagonal sums are A110508. Inverse is A110511.

Examples

			Rows begin:
1;
1,1;
3,4,1;
13,19,7,1;
67,102,44,10,1;
381,593,278,78,13,1;
From _Philippe Deléham_, Dec 01 2015: (Start)
Production matrix begins:
1, 1
2, 3, 1
2, 4, 3, 1
2, 4, 4, 3, 1
2, 4, 4, 4, 3, 1
2, 4, 4, 4, 4, 3, 1
2, 4, 4, 4, 4, 4, 3, 1
(End)

Links

Crossrefs

Cf. A000108, A052701, A084938, A110508, A110509, A110511, A114189.

Programs

  • Mathematica
    {{1}}~Join~Table[Sum[j Binomial[2 n - j - 1, n - j] Binomial[j, k] 2^(n - j), {j, 0, n}]/n, {n, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)
  • PARI
    tabl(nn)= {for (n=0, nn, for (k=0, n, if (n==0, x = 0^k, x = sum(j=0, n, j*binomial(2*n-j-1, n-j)*binomial(j, k)*2^(n-j)/n)); print1(x, ", ");); print(););} \\ Michel Marcus, Jun 18 2015

Formula

T(0,0) = 1, T(n,k) = (Sum_{j=0..n} j*C(2*n-j-1,n-j) * C(j,k) * 2^(n-j))/n.
T(n,k) = (-1)^(n-k)*A114189(n,k). - Philippe Deléham, Mar 24 2007

A110512 Expansion of (1 + x)/(1 + x + 2x^2).

Original entry on oeis.org

1, 0, -2, 2, 2, -6, 2, 10, -14, -6, 34, -22, -46, 90, 2, -182, 178, 186, -542, 170, 914, -1254, -574, 3082, -1934, -4230, 8098, 362, -16558, 15834, 17282, -48950, 14386, 83514, -112286, -54742, 279314, -169830, -388798, 728458
Offset: 0

Views

Author

Paul Barry, Jul 24 2005

Keywords

Comments

Row sums of number triangle A110511.
The sequences A110512 and A001607 are conjugated by one of the relations ((-1 + i*sqrt(7))/2)^n = a(n) + A001607(n)*(-1 + i*sqrt(7))/2 or ((-1 - i*sqrt(7))/2)^n = a(n) + A001607(n)*(-1 - i*sqrt(7))/2. These relations are connected with the Gauss sums; for example, if e := exp(i*2Pi/7) then e + e^2 + e^4 = (-1 + i*sqrt(7))/2 and e^3 + e^5 + e^6 = (-1 - i*sqrt(7))/2 -- for details see Witula's book. We also have a(n+1) = -2*A001607(n), which implies the Binet formula for a(n) (from the respective Binet formula for A001607(n) given in A001607), and A001607(n+1) = a(n) - A001607(n). - Roman Witula, Jul 27 2012
Pisano period lengths: 1, 1, 8, 1, 24, 8, 42, 1, 24, 24, 10, 8, 168, 42, 24, 2, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012

References

  • R. Witula, On some applications of formulas for unimodular complex numbers, Jacek Skalmierski's Press, Gliwice 2011 (in Polish).

Links

Programs

  • Mathematica
    CoefficientList[Series[(1 + x)/(1 + x + 2*x^2), {x,0,50}], x] (* G. C. Greubel, Aug 29 2017 *)
LinearRecurrence[{-1,-2},{1,0},40] (* Harvey P. Dale, Dec 30 2024 *)
  • PARI
    my(x='x+O('x^50)); Vec((1 + x)/(1 + x + 2*x^2)) \\ G. C. Greubel, Aug 29 2017

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-2)^(j-k)*C(k, j-k).
a(n) = (-1)^n*A078020(n). - R. J. Mathar, Feb 04 2009
a(n+2) + a(n+1) + 2*a(n) = 0. - Roman Witula, Jul 27 2012
G.f.: 2 - x + 2*x^2 + 3*x/Q(0), where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k + 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
From Ammar Khatab, Jul 11 2025: (Start)
a(n) = ((-sqrt(2))^(n+3)/sqrt(7)) * sin((n-1) * arctan(sqrt(7))).
x^n = A001607(n) * x + a(n) in Z[x]/(x^2 + x + 2).
a(n) = -2 * A001607(n-1), for n > 0. (End)

A110513 Expansion of (1 + x)/(1 + 2x + x^3).

Original entry on oeis.org

1, -1, 2, -5, 11, -24, 53, -117, 258, -569, 1255, -2768, 6105, -13465, 29698, -65501, 144467, -318632, 702765, -1549997, 3418626, -7540017, 16630031, -36678688, 80897393, -178424817, 393528322, -867954037, 1914332891, -4222194104, 9312342245, -20539017381, 45300228866, -99912799977
Offset: 0

Views

Author

Paul Barry, Jul 24 2005

Keywords

Comments

Diagonal sums of A110511.

Links

Crossrefs

A minor variation of A052980.

Programs

  • Mathematica
    CoefficientList[Series[(1+x)/(1+2x+x^3),{x,0,40}],x] (* or *) LinearRecurrence[ {-2,0,-1},{1,-1,2},40] (* Harvey P. Dale, Jun 27 2012 *)
  • PARI
    my(x='x+O('x^50)); Vec((1+x)/(1+2*x+x^3)) \\ G. C. Greubel, Aug 29 2017

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..(n-k)} (-1)^(n-k-j)*C(n-k, j)*(-2)^(j-k)*C(k, j-k).
a(0)=1, a(1)=-1, a(2)=2, a(n) = -2*a(n-1) - a(n-3). - Harvey P. Dale, Jun 27 2012

A114188 Riordan array (1/(1-x),x(1+x)/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 7, 1, 1, 16, 26, 10, 1, 1, 25, 70, 52, 13, 1, 1, 36, 155, 190, 87, 16, 1, 1, 49, 301, 553, 403, 131, 19, 1, 1, 64, 532, 1372, 1462, 736, 184, 22, 1, 1, 81, 876, 3024, 4446, 3206, 1216, 246, 25, 1, 1, 100, 1365, 6084, 11826, 11584, 6190, 1870, 317
Offset: 0

Views

Author

Paul Barry, Nov 16 2005

Keywords

Comments

Product of A007318 and A113953, that is, (1/(1-x),x/(1-x))*(1,x(1+2x)).
Row sums are A025192. Diagonal sums are A052980.
Inverse is A114189. A signed version is A110511.

Examples

			Triangle begins
1;
1, 1;
1, 4, 1;
1, 9, 7, 1;
1, 16, 26, 10, 1;
1, 25, 70, 52, 13, 1;
1, 36,155,190, 87, 16, 1;

Links

Crossrefs

Cf. A007318, A025192, A052980, A110511, A113953, A114189.

Formula

T(n, k) = Sum_{j=0..n} C(n, j)*C(k, j-k)2^(j-k).
T(n, k) = Sum_{j=0..n-k} C(k, j)*C(n+k-j, 2k).
T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)+T(n-2,k-1), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 11 2014
G.f.: 1/(1-y-x*(1+y)/(1-y)). - Vladimir Kruchinin, Apr 21 2015
Showing 1-4 of 4 results.

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