A051139 -id:A051139 - cp's OEIS frontend

A000994 Shifts 2 places left under binomial transform.

1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, 18657, 77464, 337681, 1540381, 7330418, 36301105, 186688845, 995293580, 5491595645, 31310124067, 184199228226, 1116717966103, 6968515690273, 44710457783760, 294655920067105, 1992750830574681, 13817968813639426

Offset: 0

N. J. A. Sloane

a(n) is the number of permutations of [n-1] that avoid both of the dashed patterns 1-23 and 3-12 and start with a descent (or are a singleton). For example, a(5)=5 counts 2143, 3142, 3214, 3241, 4321. - David Callan, Nov 21 2011

			A(x) = 1 + x^2/(1-x) + x^4/((1-x)^2*(1-2x)) + x^6/((1-x)^2*(1-2x)^2*(1-3x)) +...

Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 101 terms from T. D. Noe)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
N. J. A. Sloane, Transforms
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Cf. A000995, A051139, A051140. Cf. A007318.

Column k=2 of A143983. Cf. A007476, A088022, A086880.

a000994 n = a000994_list !! n
a000994_list = 1 : 0 : us where
  us = 1 : 1 : f 2 where
    f x = (1 + sum (zipWith (*) (map (a007318' x) [2..x]) us)) : f (x + 1)
-- Reinhard Zumkeller, Jun 02 2013

A000994 := proc(n) local k; option remember; if n <= 1 then 1 else 1 + add(binomial(n, k)*A000994(k - 2), k = 2 .. n); fi; end;

a[n_] := a[n] = 1 + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]; Join[{1, 0}, Table[a[n], {n, 0, 24}]] (* Jean-François Alcover, Oct 11 2011, after Maple *)

a(n)=polcoeff(sum(k=0, n, x^(2*k)*(1-k*x)/prod(j=0, k, 1-j*x+x*O(x^n))^2), n) \\ Paul D. Hanna, Nov 02 2006

Since this satisfies a recurrence similar to that of the Bell numbers (A000110), the asymptotic behavior is presumably just as complicated - see A000110 for details.
However, a(n)/A000995(n) (e.g., 77464/63117) -> 1.228..., the constant in A051148 and A051149.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n)*(1-n*x)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Nov 02 2006
Let S(n) = Sum_{k >= 1} k^n/k!^2. Then S(n) = a(n)*S(0) + A000995(n)*S(1) is stated in A086880, where S(0) = 2.279585302... (see A070910) and S(1) = 1.590636854... (see A096789). Cf. A088022. - Peter Bala, Jan 27 2015
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Aug 09 2020

A000995 Shifts left two terms under the binomial transform.

Original entry on oeis.org

0, 1, 0, 1, 2, 4, 10, 29, 90, 295, 1030, 3838, 15168, 63117, 275252, 1254801, 5968046, 29551768, 152005634, 810518729, 4472244574, 25497104007, 149993156234, 909326652914, 5674422994544, 36408092349897, 239942657880360

Offset: 0

N. J. A. Sloane

The binomial transform of A000995 has g.f. x*c(x)^2/(1+x^2*c(x)^2). - Paul Barry, Oct 06 2007
Equals row sums of triangle A137854 such that A000995(3) = 1 = first row of triangle A137854. - Gary W. Adamson, Feb 15 2008
a(n) is the number of permutations of [n-1] that avoid both of the dashed patterns 1-23 and 3-12 and start with an ascent (or are empty). For example, a(5)=4 counts 1432, 2314, 2431, 3421. - David Callan, Dec 02 2011

			A(x) = x + x^3/(1-x)^2 + x^5/((1-x)*(1-2x))^2 + x^7/((1-x)*(1-2x)*(1-3x))^2 +...

Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Cf. A000994, A051139, A051140.
Cf. A137854.
Cf. A007318.

a000995 n = a000995_list !! n
a000995_list = 0 : 1 : vs where
  vs = 0 : 1 : g 2 where
    g x = (x + sum (zipWith (*) (map (a007318' x) [2..x]) vs)) : g (x + 1)
-- Reinhard Zumkeller, Jun 02 2013

A000995 := proc(n) local k; option remember; if n <= 1 then n else n + add(binomial(n, k)*A000995(k - 2), k = 2 .. n); fi; end;

a[n_] := a[n] = If[n <= 1, n, n + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]]; Join[{0, 1}, Table[a[n], {n, 0, 24}]] (* Jean-François Alcover, May 18 2011, after Maple prog. *)
(* Computation using e.g.f.: *)
nn=20; S=(Series[-2 E^(t/2) Sqrt[E^ t] (BesselI[0, 2] BesselK[0, 2 Sqrt[E^t]] - BesselK[0, 2] Hypergeometric0F1[1, E^t]), {t, 0, nn}]); Flatten[{0, 1, FullSimplify[Table[CoefficientList[Normal[S], t][[i]] (i - 1)!, {i, 1, nn}]]}] (* Pierre-Louis Giscard, Aug 12 2014 *)

a(n)=polcoeff(sum(k=0,n,x^(2*k+1)/prod(j=0,k,1-j*x+x*O(x^n))^2),n) \\ Paul D. Hanna, Oct 28 2006

Since this satisfies a recurrence similar to that of the Bell numbers (A000110), the asymptotic behavior is presumably just as complicated - see A000110 for details.
However, A000994(n)/A000995(n) [ e.g., 77464/63117 ] -> 1.228..., the constant in A051148 and A051149.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n+1)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Oct 28 2006
G.f.: (1+2*x^2*c(x)^2)/(1+x^2*c(x^2)), c(x) the g.f. of A000108. - Paul Barry, Oct 06 2007. This g.f. is incorrect. - Vaclav Kotesovec, Aug 14 2014
E.g.f: -2 * exp(x) *( BesselI_0(2) * BesselK_0(2*exp(x/2)) - BesselK_0(2) * 0F1([], [1], exp(x)) ); see the Mathematica program. - Pierre-Louis Giscard, Aug 12 2014
G.f. A(x) satisfies: A(x) = x*(1 + x*A(x/(1 - x))/(1 - x)). - Ilya Gutkovskiy, May 02 2019

More terms from Paul D. Hanna, Oct 28 2006

A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2x)) / (1 + 2x).

Original entry on oeis.org

1, 1, 1, -1, 1, -3, 17, -85, 385, -1767, 8929, -50633, 312705, -2036267, 13794417, -97295069, 717808897, -5549714767, 44868094145, -377741383697, 3298933836033, -29813463964115, 278462029910993, -2685972391332837, 26733375327601281, -274247228584531767

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

sign

Shifts 2 places left under 2nd-order inverse binomial transform.

Cf. A007472, A009235, A010739, A051139, A351184, A351185, A351186, A351187.

nmax = 25; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 2 x)]/(1 + 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-2)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-2)^k * a(n-k-2).

A351184 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 3x)) / (1 + 3x).

Original entry on oeis.org

1, 1, 1, -2, 4, -11, 55, -359, 2359, -15230, 100840, -716555, 5580145, -47230091, 425472229, -4013326982, 39379161136, -402010392971, 4279164575167, -47533936734179, 550239127112107, -6618018093867506, 82447377648018700, -1061324336149876667, 14095604842846277617

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

sign

Shifts 2 places left under 3rd-order inverse binomial transform.

Cf. A010739, A051139, A317996, A350456, A351049, A351185, A351186, A351187.

nmax = 24; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 3 x)]/(1 + 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-3)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-3)^k * a(n-k-2).

A351185 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 4x)) / (1 + 4x).

Original entry on oeis.org

1, 1, 1, -3, 9, -31, 153, -1075, 8689, -72031, 605201, -5282051, 49239225, -497094079, 5410919273, -62597718643, 759331611489, -9586004915007, 125701843190689, -1713676634245251, 24313707650733289, -358906747784541151, 5502327502961296825, -87382907614533531443

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

sign

Shifts 2 places left under 4th-order inverse binomial transform.

Cf. A010739, A051139, A318179, A350456, A351050, A351184, A351186, A351187.

nmax = 23; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 4 x)]/(1 + 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-4)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 23}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-4)^k * a(n-k-2).

A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5x)) / (1 + 5x).

Original entry on oeis.org

1, 1, 1, -4, 16, -69, 371, -2719, 24691, -243804, 2479276, -25931249, 284075601, -3320433179, 41744590941, -561939568544, 8008026088996, -119496752915869, 1854697111334891, -29870689367146379, 499291484226079551, -8668202648905259624, 156301404533216141576

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

sign

Shifts 2 places left under 5th-order inverse binomial transform.

Cf. A010739, A051139, A318180, A350456, A351056, A351184, A351185, A351187.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 5 x)]/(1 + 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-5)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-5)^k * a(n-k-2).

A351187 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 6x)) / (1 + 6x).

Original entry on oeis.org

1, 1, 1, -5, 25, -131, 793, -6137, 60049, -670919, 7930321, -96775853, 1225237609, -16333089227, 232150489129, -3531321746465, 57178717416097, -975918663642767, 17400776511175201, -322309002081819221, 6188520430773389881, -123166171374344928275, 2542231599282355411897

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

sign

Shifts 2 places left under 6th-order inverse binomial transform.

Cf. A010739, A051139, A318181, A350456, A351057, A351184, A351185, A351186.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 6 x)]/(1 + 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-6)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-6)^k * a(n-k-2).

cp's OEIS Frontend

A000994 Shifts 2 places left under binomial transform.

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A000995 Shifts left two terms under the binomial transform.

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A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2*x)) / (1 + 2*x).

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A351184 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 3*x)) / (1 + 3*x).

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A351185 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 4*x)) / (1 + 4*x).

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A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5*x)) / (1 + 5*x).

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A351187 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 6*x)) / (1 + 6*x).

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A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2x)) / (1 + 2x).

A351184 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 3x)) / (1 + 3x).

A351185 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 4x)) / (1 + 4x).

A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5x)) / (1 + 5x).

A351187 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 6x)) / (1 + 6x).