A055877 -id:A055877 - cp's OEIS frontend

A055878 Least positive sequence with Hankel transform {1,1,1,1,1,...}.

1, 1, 2, 1, 6, 1, 43, 1, 7794, 1, 387355897233, 1, 58111569344245781528237519282283636, 1

Offset: 1

John W. Layman, Jul 15 2000

nonn

Hankel transform {t(n)} of {a(n)} is given by t(n) = Det[{a(1), a(2), ..., a(n)}, {a(2), a(3), ..., a(n+1)}, ..., {a(n), a(n+1), ..., a(2n-1)}].

M M Mangontarum, O I Cauntongan, A P Macodi-Ringia, The Noncentral Version of the Whitney Numbers: A Comprehensive Study, International Journal of Mathematics and Mathematical Sciences, Volume 2016, Article ID 6206207, 16 pages; http://dx.doi.org/10.1155/2016/6206207. See reference 21.

M. Garcia Armas, B. A. Sethuraman, A note on the Hankel Transform of the Central Binomial Coefficients, JIS 11 (2008) 08.5.8
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A055877, A055564, A055879.

Next term is too large to include.

A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2n+1)] are > 0.

Original entry on oeis.org

1, 2, 5, 13, 35, 98, 287, 883, 2858, 9708, 34411, 126337, 476767, 1836851, 7185420, 28420613, 113317776, 454468077, 1830556209, 7397188271, 29965426959, 121620119888, 494365414071, 2011965781648, 8196475452837, 33419092543257, 136353532725534, 556669705441210

Offset: 0

Thomas Scheuerle, Sep 23 2024

A Stieltjes moment sequence by its definition.
The Hankel sequence transform gives {1, 1, 1, 1, 1, ...}.
The definition causes that the Hankel sequence transform starting with the second term of this sequence becomes {2, 1, 1, 1, ...}. This single exceptional 2 causes high complexity in the generating function and makes a nice combinatorial interpretation less likely, therefore the keyword "less" was considered.

Cf. A000108 (We obtain the Catalan numbers if we use "least positive sequence" in the definition instead of "least increasing").
Cf. A375181 (Binomial transform).
Cf. A034807, A011973.
Cf. also A055877, A055878, A055879, A350349.

hankelok(s) = {my(m1=floor((#s+1)/2)); my(m2=floor(#s/2)); my(h1=matrix(m1,m1,x,y,s[x+y-1]));  my(h2=matrix(m2,m2,x,y,s[x+y])); return((matdet(h1) > 0) && (matdet(h2) > 0))}
a(max_n) = {my(s=[1,2],k=3); while(#s < max_n, while(hankelok(concat(s,[k]))==0,k=k+1); s=concat(s,[k])); return(s)}

my(N=30, x='x+O('x^N)); Vec(1/(1-2*x/(1-(1/2)*x/(1-(1/2)*x/(1-2*x/(1-((1-sqrt(1-4*x))/(2*x))*x))))))

a(n) = if(n<3, [1, 2, 5][n+1], sum(k=1, floor((n+1)/2), (binomial(n-k+1, k)+binomial(n-k, k-1)-binomial(n-k-3, k-4))*(-1)^(k+1)*a(n-k)))

G.f.: 1/(1-2*x/(1-(1/2)*x/(1-(1/2)*x/(1-2*x/(1-C(x)*x))))), C(x) is the generating function of the Catalan numbers.
G.f.: (1 - sqrt(1 - 4*x)*(-1 + x) - 5*x + 2*x^2)/(1 - 7*x + 11*x^2 + sqrt(1 - 4*x)*(1 - 3*x + x^2)).
(sqrt((x - 4)/x) + 2*x*(13 + (x - 7)*x) - 9)/(2*((x - 4)*(x - 3)*(x - 2)*x - 1)) = Sum_{k>=0} a(k)/x^(k+1).
a(n) = Sum_{k=1..floor((n+1)/2)} (binomial(n-k+1, k) + binomial(n-k, k-1) - binomial(n-k-3, k-4))*(-1)^(k+1)*a(n-k), for n >= 3.
a(n) = Sum_{k=1..floor((n+1)/2)} (A034807(n+1, k) - A011973(n+1, k-4))*(-1)^(k+1)*a(n-k), for n >= 3.

cp's OEIS Frontend

A055878 Least positive sequence with Hankel transform {1,1,1,1,1,...}.

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A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2n+1)] are > 0.

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cp's OEIS Frontend

A055878 Least positive sequence with Hankel transform {1,1,1,1,1,...}.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2*n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2*n+1)] are > 0.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

PARI

PARI

Formula

A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2n+1)] are > 0.