A000995 -id:A000995 - cp's OEIS frontend

A051139 a(n) = A000994(n+2) - A000995(n+2).

1, 0, 0, 1, 3, 7, 19, 64, 236, 893, 3489, 14347, 62429, 285580, 1362372, 6749337, 34683211, 184774851, 1019351071, 5813020060, 34206071992, 207391313189, 1294092695729, 8302365433863, 54713262186745, 370027343460584, 2565874205681368

Offset: 0

N. J. A. Sloane

			a(7) = 359 - 295 = 64.

Reinhard Zumkeller, Table of n, a(n) for n = 0..100

Cf. A000994, A000995.

a051139 n = a000994 (n + 2) - a000995 (n + 2)
-- Reinhard Zumkeller, Jun 02 2013

a4[?Negative] = 0; a4[n] := a4[n] = 1 + Sum[ Binomial[n, k]*a4[k-2], {k, 2, n}]; a5[?Negative] = 0; a5[n] := a5[n] = n + Sum[ Binomial[n, k]*a5[k-2], {k, 2, n}]; a[n_] := a4[n] - a5[n]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 11 2013 *)

For recurrence see A000994 and A000995.

A007476 Shifts 2 places left under binomial transform.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 23, 65, 199, 654, 2296, 8569, 33825, 140581, 612933, 2795182, 13298464, 65852873, 338694479, 1805812309, 9963840219, 56807228074, 334192384460, 2026044619017, 12642938684817, 81118550133657, 534598577947465, 3615474317688778, 25070063421597484

Offset: 0

N. J. A. Sloane

Starting (1, 2, 4, 9, 23, ...) = row sums of triangle A153859. - Gary W. Adamson, Jan 02 2009
Binomial transform of the sequence starting (1, 1, 2, 4, 9, ...) = first differences of (1, 2, 4, 9, 23, ...); that is, (1, 2, 5, 14, 42, 134, 455, 1642, ...). - Gary W. Adamson, May 20 2013
Row sums of triangle A256161. - Margaret A. Readdy, Mar 16 2015
RG-words corresponding to set partitions of {1, ..., n} with every even entry appearing exactly once. - Margaret A. Readdy, Mar 16 2015
a(n) is the number of partitions of [n] whose blocks can be written such that the smallest elements form an increasing sequence and the largest elements form a decreasing sequence. a(5) = 9: 12345, 1235|4, 1245|3, 125|34, 1345|2, 135|24, 145|23, 15|234, 15|24|3. - Alois P. Heinz, Apr 24 2023

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, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
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]
Yue Cai and Margaret Readdy, Negative q-Stirling numbers, arXiv:1506.03249 [math.CO], 2015.
A. Claesson and T. Mansour, Permutations avoiding a pair of Babson-Steingrimsson patterns, arXiv:math/0107044 [math.CO], 2001-2010.
Rigoberto Flórez, José L. Ramírez, Fabio A. Velandia, and Diego Villamizar, Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions, arXiv:2308.02059 [math.CO], 2023. See Table 1 p. 13.
N. J. A. Sloane, Transforms
L. Sze, OEIS conjecture 70

Cf. A000994, A000995, A092920, A153859.

Row sums of A246118.

a:= proc(n) option remember; `if`(n<2, 1,
      add(a(j)*binomial(n-2, j), j=0..n-2))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Jul 29 2019

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n-2, k] a[k], {k, 0, n-2}]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 08 2012, after Ralf Stephan *)

a(n)=if(n<2, 1, sum(k=0, n-2, binomial(n-2, k)*a(k))) /* Ralf Stephan; corrected by Manuel Blum, May 22 2010 */

G.f.: Sum_{k>=0} x^(2k)/(Product_{m=0..k-1} (1-mx) * Product_{m=0..k+1} (1-mx)).
G.f. A(x) satisfies A(x) = 1 + x + (x^2/(1-x))*A(x/(1-x)). - Vladimir Kruchinin, Nov 28 2011
a(n) = A000994(n) + A000995(n). - Peter Bala, Jan 27 2015

Spelling correction by Jason G. Wurtzel, Aug 22 2010

A000994 Shifts 2 places left under binomial transform.

Original entry on oeis.org

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

A086880 a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers.

Original entry on oeis.org

2, 1, 2, 3, 7, 17, 45, 128, 391, 1287, 4524, 16889, 66657, 276982, 1207598, 5507362, 26203307, 129757596, 667358910, 3558097578, 19632277761, 111930731957, 658482495614, 3992062349412, 24911272290567, 159833355923362

Offset: 0

Paul D. Hanna, Sep 16 2003

nonn

Define B(n) = sum(k=0, infinity, k^n/(k!)^2), then there exists a complex linear relation: B(3) = B(2) + B(1); B(4) = 2*B(3); B(5) = 2*B(4) + B(2); B(6) = 5*B(4) + 3*B(2); B(7) = 7*B(5) + B(3); B(12) = B(11) + 11*B(10); ...

			a(5) = floor(1^5/(1!)^2 + 2^5/(2!)^2 + 3^5/(3!)^2 + 4^5/(4!)^2 +...)

Vaclav Kotesovec, Table of n, a(n) for n = 0..646

Cf. A000994, A000995, A006789.

Table[Floor[Sum[k^n/(k!)^2,{k,0,Infinity}]],{n,0,20}] (* Vaclav Kotesovec, Jul 31 2014 *)
Flatten[{2, 1, Table[Floor[HypergeometricPFQ[ConstantArray[2, n-2], ConstantArray[1, n-1], 1]], {n,2,20}]}] (* Vaclav Kotesovec, May 23 2015 *)

sum(k>=0, k^n/(k!)^2) = A000994(n)*BesselI(0, 2) + A000995(n)*BesselI(1, 2), using Bessel function values BesselI(0, 2)=2.2795853023..., BesselI(1, 2) = 1.5906368546... (A096789) and where A000994 and A000995 shift 2 places left under binomial transform: A000994={1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, ...} A000995={0, 1, 0, 1, 2, 4, 10, 29, 90, 295, 1030, 3838, ...}.

A000997 From a differential equation.

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 3, 5, 12, 36, 110, 326, 963, 2964, 9797, 34818, 130585, 506996, 2018454, 8238737, 34627390, 150485325, 677033911, 3147372610, 15066340824, 74025698886, 372557932434, 1919196902205, 10119758506626, 54627382038761, 301832813494746

Offset: 0

N. J. A. Sloane

Shifts 3 places left under binomial transform. - Olivier Gérard, Aug 12 2016

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).

S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Cf. A000995.

a := proc(n) option remember; local k; if n<=2 then [0, 1, 0][n+1] else add (binomial(n-3, k)*a(k), k=1..n-3) fi end: seq(a(n), n=0..29); # Sean A. Irvine, Mar 27 2015

m = 30; A[_] = 0;
Do[A[x_] = x (1 + x^2 A[x/(1 - x)]/(1 - x)) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 23 2019 *)

G.f. A(x) satisfies: A(x) = x*(1 + x^2*A(x/(1 - x))/(1 - x)). - Ilya Gutkovskiy, May 02 2019

More terms from Sean A. Irvine, Mar 27 2015

A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

0, 1, 0, 1, 4, 13, 44, 173, 792, 4009, 21608, 122761, 737340, 4696341, 31665076, 224846037, 1672266352, 12976252561, 104816144656, 880061135057, 7670326372532, 69286959112797, 647568753568636, 6251768635591613, 62255057942504968, 638658964709824185

Offset: 0

Ilya Gutkovskiy, Feb 03 2022

nonn

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

Cf. A000995, A004211, A007472, A351053, A351128, A351132, A351143, A351161.

bintr:= proc(p) local b;
          b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
        end:
b:= (bintr@@2)(a):
a:= n-> `if`(n<2, n, b(n-2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 07 2025

nmax = 25; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 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}];
(* another pprogram *)
B[x_] := BesselK[0, 1]*BesselI[0, Exp[x]] - BesselI[0, 1]*BesselK[0, Exp[x]];
a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!;
Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)

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

E.g.f.: BesselK(0, 1)*BesselI(0, exp(x)) - BesselI(0, 1)*BesselK(0, exp(x)). - Ven Popov, Apr 25 2025

A051148 Decimal expansion of BesselK(1,2)/BesselK(0,2).

Original entry on oeis.org

1, 2, 2, 8, 0, 3, 6, 9, 2, 9, 8, 1, 8, 9, 0, 7, 9, 7, 5, 7, 4, 2, 6, 7, 2, 4, 5, 2, 0, 8, 1, 7, 1, 8, 9, 0, 4, 2, 3, 9, 3, 6, 6, 9, 5, 2, 6, 6, 0, 8, 5, 9, 6, 9, 4, 4, 6, 7, 0, 3, 8, 1, 4, 9, 1, 3, 7, 4, 7, 1, 9, 9, 1, 5, 7, 0, 3, 8, 9, 8, 3, 9, 7, 0, 5, 3, 8, 1, 5, 2, 4, 7, 3, 8, 2, 2, 9, 2, 1

Offset: 1

N. J. A. Sloane, E. M. Rains

			1.228036929818907975742672452...

Harry J. Smith, Table of n, a(n) for n = 1..2000
Index entries for sequences related to Bessel functions or polynomials

Cf. A051149, A000994, A000995.

RealDigits[ BesselK[1, 2] / BesselK[0, 2], 10, 99] // First (* Jean-François Alcover, Mar 07 2013 *)

{ allocatemem(932245000); default(realprecision, 2080); x=besselk(1,2)/besselk(0,2); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b051148.txt", n, " ", d)); } \\ Harry J. Smith, Apr 29 2009

Fixed my PARI program, had -n. - Harry J. Smith, May 19 2009

A051149 Continued fraction for BesselK(1,2)/BesselK(0,2).

Original entry on oeis.org

1, 4, 2, 1, 1, 2, 8, 1, 6, 1, 4, 2, 1, 8, 1, 1, 7, 1, 6, 4, 2, 1, 6, 1, 1, 2, 1, 1, 1, 2, 5, 1, 3, 3, 1, 3, 1, 8, 1, 1, 1, 1, 1, 2, 3, 8, 3, 29, 1, 7, 1, 57, 1, 121, 2, 14, 2, 8, 1, 1, 16, 1, 3, 1, 5, 1, 5, 1, 4, 17, 1, 5, 6, 1, 3, 2, 9, 7, 1, 4, 4, 1, 1, 16, 3, 5, 2, 1, 2, 1

Offset: 0

N. J. A. Sloane, E. M. Rains

			1.228036929818907975742672452...
BesselK(1,2)/BesselK(0,2) = 1 + 1/(4 + 1/(2 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 29 2009

Harry J. Smith, Table of n, a(n) for n = 0..2000
G. Xiao, Contfrac
Index entries for sequences related to Bessel functions or polynomials
Index entries for continued fractions for constants

Cf. A051148, A000994, A000995.

ContinuedFraction[BesselK[1,2]/BesselK[0,2],90] (* Harvey P. Dale, Sep 07 2018 *)

{ allocatemem(932245000); default(realprecision, 2100); x=contfrac(besselk(1,2)/besselk(0,2)); for (n=1, 2001, write("b051149.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 29 2009

A351053 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 3x)) / (1 - 3x).

Original entry on oeis.org

0, 1, 0, 1, 6, 28, 126, 613, 3438, 22159, 157362, 1189126, 9436320, 78690781, 692478684, 6439539457, 63106488618, 648453907216, 6952719052134, 77521908188737, 897132401326458, 10764085132255807, 133774484448519294, 1720018195807299418, 22847325911461934352

Offset: 0

Ilya Gutkovskiy, Feb 03 2022

nonn

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

Cf. A000995, A004212, A351028, A351049, A351128, A351132, A351144, A351161.

nmax = 24; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 3 x)]/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 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) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 3^k * a(n-k-2).

A351128 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 4x)) / (1 - 4x).

Original entry on oeis.org

0, 1, 0, 1, 8, 49, 280, 1649, 10800, 81505, 696400, 6472033, 63562872, 652984977, 7026210728, 79547049681, 949709767904, 11936248012993, 157219119485216, 2159448120457409, 30811324011852136, 455635009201780977, 6975424580445456056, 110478282815356437809

Offset: 0

Ilya Gutkovskiy, Feb 03 2022

nonn

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

Cf. A000995, A004213, A351028, A351050, A351053, A351132, A351150, A351161.

nmax = 23; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 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) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 4^k * a(n-k-2).

cp's OEIS Frontend

A051139 a(n) = A000994(n+2) - A000995(n+2).

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A007476 Shifts 2 places left under binomial transform.

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A000994 Shifts 2 places left under binomial transform.

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A086880 a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers.

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A000997 From a differential equation.

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A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).

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A051148 Decimal expansion of BesselK(1,2)/BesselK(0,2).

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A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2x)) / (1 - 2x).

A351053 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 3x)) / (1 - 3x).

A351128 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 4x)) / (1 - 4x).