A105438 -id:A105438 - cp's OEIS frontend

A105522 Inverse of number triangle A105438.

1, -2, 1, 1, -2, 1, 2, 0, -2, 1, -2, 4, -1, -2, 1, -4, -1, 6, -2, -2, 1, 5, -10, 1, 8, -3, -2, 1, 10, 4, -18, 4, 10, -4, -2, 1, -14, 28, 0, -28, 8, 12, -5, -2, 1, -28, -14, 56, -8, -40, 13, 14, -6, -2, 1, 42, -84, -6, 96, -21, -54, 19, 16, -7, -2, 1, 84, 48, -180, 15, 150, -40, -70, 26, 18, -8, -2, 1, -132, 264, 33, -330, 55, 220, -66

Offset: 0

Paul Barry, Apr 11 2005

Row sums A105523 have g.f. 1-xc(-x^2) where c(x) is the g.f. of A000108. Diagonal sums are A105524.

			Rows begin {1}, {-2, 1}, {1, -2, 1}, {2, 0, -2, 1}, {-2, 4, -1, -2, 1}, ...

Riordan array ((1+2x+4x^2-(1+2x)sqrt(1+4x^2))/(2x^2), (sqrt(1+4x^2)-1)/(2x))

A105523 Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, -1, 0, 1, 0, -2, 0, 5, 0, -14, 0, 42, 0, -132, 0, 429, 0, -1430, 0, 4862, 0, -16796, 0, 58786, 0, -208012, 0, 742900, 0, -2674440, 0, 9694845, 0, -35357670, 0, 129644790, 0, -477638700, 0, 1767263190, 0

Offset: 0

Paul Barry, Apr 11 2005

Row sums of A105522. Row sums of inverse of A105438.

First column of number triangle A106180.

			G.f. = 1 - x + x^3 - 2*x^5 + 5*x^7 - 14*x^9 + 42*x^11 - 132*x^13 + 429*x^15 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 313.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.

Cf. A000108, A097331, A090192, A090181, A105522, A105438.

m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 + 2*x - Sqrt(1+4*x^2))/(2*x))); // G. C. Greubel, Sep 16 2018

A105523_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w]:=-a[w-1]+(-1)^w*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a,list)end: A105523_list(40); # Peter Luschny, May 19 2011

a[n_?EvenQ] := 0; a[n_?OddQ] := 4^n*Gamma[n/2] / (Gamma[-n/2]*(n+1)!); a[0] = 1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
CoefficientList[Series[(1 + 2 x - Sqrt[1 + 4 x^2])/(2 x), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 01 2014 *)
a[ n_] := SeriesCoefficient[ (1 + 2 x - Sqrt[ 1 + 4 x^2]) / (2 x), {x, 0, n}]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], a[n] = -2 a[n - 1] + Sum[ a[j] a[n - j - 1], {j, 0, n - 1}]]; (* Michael Somos, Jun 17 2015 *)

{a(n) = local(A); if( n<0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = -2 * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

def A105523(n):
    if is_even(n): return 0 if n>0 else 1
    return -(sqrt(pi)*2^(n-1))/(gamma(1-n/2)*gamma((n+3)/2))
[A105523(n) for n in (0..29)] # Peter Luschny, Oct 31 2014

G.f.: (1 + 2*x - sqrt(1+4*x^2))/(2*x).
a(n) = 0^n + sin(Pi*(n-2)/2)(C((n-1)/2)(1-(-1)^n)/2).
G.f.: 1/(1+x/(1-x/(1+x/(1-x/(1+x/(1-x.... (continued fraction). - Paul Barry, Jan 15 2009
a(n) = Sum{k = 0..n} A090181(n,k)*(-1)^k. - Philippe Deléham, Feb 02 2009
a(n) = (1/n)*sum_{i = 0..n-1} (-2)^i*binomial(n, i)*binomial(2*n-i-2, n-1). - Vladimir Kruchinin, Dec 26 2010
With offset 1, a(n) = -2 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k), for n>1. - Michael Somos, Jul 25 2011
D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 18 2014
For nonzero terms, a(n) ~ (-1)^((n+1)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - Fung Lam, Mar 17 2014
a(n) = -(sqrt(Pi)*2^(n-1))/(Gamma(1-n/2)*Gamma((n+3)/2)) for n odd. - Peter Luschny, Oct 31 2014
From Peter Bala, Apr 20 2024: (Start)
a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n + k, 2*k)*Catalan(k), where Catalan(k) = A000108(k).
a(n) = (-2)^n * hypergeom([-n, n+1], [2], 1/2).
O.g.f.: A(x) = 1/x * series reversion of x*(1 - x)/(1 - 2*x).  Cf. A152681. (End)

Typo in definition corrected by Robert Israel, Oct 31 2014

A221074 Simple continued fraction expansion of product {n >= 0} {1 - sqrt(m)[sqrt(m) - sqrt(m-1)]^(4n+3)}/{1 - sqrt(m)[sqrt(m) - sqrt(m-1)]^(4n+1)} at m = 3.

Original entry on oeis.org

2, 8, 1, 16, 1, 96, 1, 176, 1, 968, 1, 1760, 1, 9600, 1, 17440, 1, 95048, 1, 172656, 1, 940896, 1, 1709136, 1, 9313928, 1, 16918720, 1, 92198400, 1, 167478080, 1, 912670088, 1, 1657862096, 1

Offset: 0

Peter Bala, Jan 06 2013

Simple continued fraction expansion of product {n >= 0} {1 - sqrt(m)*[sqrt(m) - sqrt(m-1)]^(4*n+3)}/{1 - sqrt(m)*[sqrt(m) - sqrt(m-1)]^(4*n+1)} at m = 3. For other cases see A221073 (m = 2), A221075 (m = 4) and A221076 (m = 5).

			Product {n >= 0} {1 - sqrt(3)*(sqrt(3) - sqrt(2))^(4*n+3)}/{1 - sqrt(3)*(sqrt(3) - sqrt(2))^(4*n+1)} = 2.11180 16361 44098 52896 ...
= 2 + 1/(8 + 1/(1 + 1/(16 + 1/(1 + 1/(96 + 1/(1 + 1/(176 + ...))))))).
Since (sqrt(3) - sqrt(2))^3 = 9*sqrt(3) - 11*sqrt(2) we have the following simple continued fraction expansion:
product {n >= 0} {1 - sqrt(3)*(9*sqrt(3) - 11*sqrt(2))^(4*n+3)}/{1 - sqrt(3)*(9*sqrt(3) - 11*sqrt(2))^(4*n+1)} = 1 + 1/(16 + 1/(1 + 1/(968 + 1/(1 + 1/(17440 + 1/(1 + 1/(940896 + ...))))))).

Peter Bala, Some simple continued fraction expansions for an infinite product, Part 1
Index entries for linear recurrences with constant coefficients, signature (0,1,0,10,0,-10,0,-1,0,1).

Cf. A098297, A105438, A132596, A174500, A221073 (m = 2), A221075 (m = 4), A221076 (m = 5).

a(2*n) = 1 for n >= 1. For n >= 1 we have
a(4*n - 3) = (sqrt(3) + sqrt(2))^(2*n) + (sqrt(3) - sqrt(2))^(2*n) - 2;
a(4*n - 1) = 1/sqrt(3)*{(sqrt(3) + sqrt(2))^(2*n + 1) + (sqrt(3) - sqrt(2))^(2*n + 1)} - 2.
a(4*n - 3) = 8*A098297(n) = 4*A132596(n); a(4*n - 1) = 4*A105038(n).
O.g.f.: 2 + x^2/(1 - x^2) + 8*x*(1 + x^2)^2/(1 - 11*x^4 + 11*x^8 - x^12) = 2 + 8*x + x^2 + 16*x^3 + x^4 + 96*x^5 + ....
If we denote the present sequence by [2; 8, 1, c(3), 1, c(4), 1, ...] then for k >= 1 the sequence [1; c(2*k+1), 1, c(2*(2*k+1)), 1, c(3*(2*k+1)), 1, ...] gives the simple continued fraction expansion of product {n >= 0} [1-sqrt(3)*{(sqrt(3)-sqrt(2))^(2*k+1)}^(4*n+3)]/[1 - sqrt(3)*{(sqrt(3)-sqrt(2))^(2*k+1)}^(4*n+1)]. An example is given below.
O.g.f.: (x^10-2*x^8-10*x^6+20*x^4-8*x^3+x^2-8*x-2) / ((x-1)*(x+1)*(x^8-10*x^4+1)). - Colin Barker, Jan 10 2014

A345123 Number T(n,k) of ordered subsequences of {1,...,n} containing at least k elements and such that the first differences contain only odd numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 7, 6, 3, 1, 12, 11, 7, 3, 1, 20, 19, 14, 8, 3, 1, 33, 32, 26, 17, 9, 3, 1, 54, 53, 46, 34, 20, 10, 3, 1, 88, 87, 79, 63, 43, 23, 11, 3, 1, 143, 142, 133, 113, 83, 53, 26, 12, 3, 1, 232, 231, 221, 196, 156, 106, 64, 29, 13, 3, 1, 376, 375, 364, 334, 279, 209, 132, 76, 32, 14, 3, 1

Offset: 0

Alois P. Heinz, Jun 08 2021

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2k if k >= 2 and of order 3 for k in {0, 1}.

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,2) = 1: [1,2].
T(3,1) = 6: [1], [2], [3], [1,2], [2,3], [1,2,3].
T(4,0) = 12: [], [1], [2], [3], [4], [1,2], [1,4], [2,3], [3,4], [1,2,3], [2,3,4], [1,2,3,4].
T(6,3) = 17: [1,2,3], [1,2,5], [1,4,5], [2,3,4], [2,3,6], [2,5,6], [3,4,5], [4,5,6], [1,2,3,4], [1,2,3,6], [1,2,5,6], [1,4,5,6], [2,3,4,5], [3,4,5,6], [1,2,3,4,5], [2,3,4,5,6], [1,2,3,4,5,6].
Triangle T(n,k) begins:
    1;
    2,   1;
    4,   3,   1;
    7,   6,   3,   1;
   12,  11,   7,   3,   1;
   20,  19,  14,   8,   3,   1;
   33,  32,  26,  17,   9,   3,  1;
   54,  53,  46,  34,  20,  10,  3,  1;
   88,  87,  79,  63,  43,  23, 11,  3,  1;
  143, 142, 133, 113,  83,  53, 26, 12,  3, 1;
  232, 231, 221, 196, 156, 106, 64, 29, 13, 3, 1;
  ...

Chu, Hung Viet, Various Sequences from Counting Subsets, Fib. Quart., 59:2 (May 2021), 150-157.

Alois P. Heinz, Rows n = 0..140, flattened
Chu, Hung Viet, Various Sequences from Counting Subsets, arXiv:2005.10081 [math.CO], 2021.

Columns k=0-3 give: A000071(n+3), A001911, A001924(n-1), A344004.
T(2n,n) give A340766.
Cf. A073044, A105438.

b:= proc(n, l, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(0
      in [l, irem(1+l-n, 2)], b(n-1, n, max(0, t-1)), 0)+b(n-1, l, t))
    end:
T:= (n, k)-> b(n, 0, k):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
g:= proc(n, k) option remember; `if`(k>n, 0,
     `if`(k in [0, 1], n^k, g(n-1, k-1)+g(n-2, k)))
    end:
T:= proc(n, k) option remember;
     `if`(k>n, 0, g(n, k)+T(n, k+1))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);
# third Maple program:
T:= proc(n, k) option remember; `if`(k>n, 0, binomial(iquo(n+k, 2), k)+
      `if`(k>0, binomial(iquo(n+k-1, 2), k), 0)+T(n, k+1))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);

T[n_, k_] := T[n, k] = If[k > n, 0, Binomial[Quotient[n+k, 2], k] +
     If[k > 0, Binomial[Quotient[n+k-1, 2], k], 0] + T[n, k+1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 3rd Maple program *)

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A105522 Inverse of number triangle A105438.

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A105523 Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.

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A221074 Simple continued fraction expansion of product {n >= 0} {1 - sqrt(m)*[sqrt(m) - sqrt(m-1)]^(4*n+3)}/{1 - sqrt(m)*[sqrt(m) - sqrt(m-1)]^(4*n+1)} at m = 3.

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A345123 Number T(n,k) of ordered subsequences of {1,...,n} containing at least k elements and such that the first differences contain only odd numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

Mathematica

A221074 Simple continued fraction expansion of product {n >= 0} {1 - sqrt(m)[sqrt(m) - sqrt(m-1)]^(4n+3)}/{1 - sqrt(m)[sqrt(m) - sqrt(m-1)]^(4n+1)} at m = 3.