A000994 -id:A000994 - cp's OEIS frontend

A351151 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 5x)) / (1 - 5x).

1, 0, 1, 5, 26, 145, 901, 6420, 52501, 480955, 4795626, 51066375, 576182001, 6879462680, 86955722401, 1162559359745, 16392133866026, 242734091500445, 3758825675820501, 60660434188558780, 1017770666417312501, 17725289455315892375, 320047193447632729626

Offset: 0

Ilya Gutkovskiy, Feb 02 2022

nonn

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

Cf. A000994, A005011, A351056, A351143, A351144, A351150, A351152.

nmax = 22; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; 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) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A351152 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6x)) / (1 - 6x).

Original entry on oeis.org

1, 0, 1, 6, 37, 240, 1693, 13446, 122329, 1261104, 14332681, 175123446, 2267871517, 30981705984, 446571784261, 6798161166486, 109220619908593, 1846729159654560, 32726973173941585, 605358657750562470, 11648701234354836565, 232655173657593759312

Offset: 0

Ilya Gutkovskiy, Feb 02 2022

nonn

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

Cf. A000994, A005012, A351057, A351143, A351144, A351150, A351151.

nmax = 21; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 21}]

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

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

A336970 G.f. A(x) satisfies: A(x) = 1 - x^2 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 0, -1, -1, 0, 3, 9, 16, 7, -87, -472, -1567, -3375, -216, 45927, 308107, 1372744, 4351599, 5711849, -49345432, -547773585, -3517370859, -16914970464, -56474505155, -25470754271, 1593389360016, 17323737305039, 125785962635543, 706598399399184

Offset: 0

Ilya Gutkovskiy, Aug 09 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..657

Cf. A000587, A000994, A336971.

nmax = 28; A[] = 0; Do[A[x] = 1 - x^2 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]

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

A246118 T(n,k), for n,k >= 1, is the number of partitions of the set [n] into k blocks, where, if the blocks are arranged in order of their minimal element, the odd-indexed blocks are all singletons.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 11, 6, 1, 0, 1, 5, 26, 23, 9, 1, 0, 1, 6, 57, 72, 50, 12, 1, 0, 1, 7, 120, 201, 222, 86, 16, 1, 0, 1, 8, 247, 522, 867, 480, 150, 20, 1, 0, 1, 9, 502, 1291, 3123, 2307, 1080, 230, 25, 1, 0, 1, 10, 1013, 3084, 10660, 10044, 6627, 2000, 355, 30, 1

Offset: 1

Peter Bala, Aug 14 2014

Unsigned matrix inverse of A246117. Analog of the Stirling numbers of the second kind, A048993.
This is the triangle of connection constants between the monomial polynomials x^n and the polynomial sequence [x, x^2, x^2*(x - 1), x^2*(x - 1)^2, x^2*(x - 1)^2*(x - 2), x^2*(x - 1)^2*(x - 2)^2, ...]. An example is given below.
Except for differences in offset, this triangle is the Galton array G(floor(k/2),1) in the notation of Neuwirth with inverse array G(-floor(n/2),1).
Essentially the same as A256161. - Peter Bala, Apr 14 2018
From Peter Bala, Feb 10 2020: (Start)
The sums S(n):= Sum_{k >= 0} k^n*(x^k/k!)^2, n = 2,3,4,..., can be expressed as a linear combination of the sums S(0) and S(1) with polynomial coefficients, namely, S(n) = E(n,x)*S(0) + (1/x)*O(n,x)* S(1,x), where E(n,x) = Sum_{k >= 1} T(n,2*k)*x^(2*k) and O(n,x) = Sum_{k >= 0} T(n,2*k+1)*x^(2*k+1) are the even and odd parts of the n-th row polynomial of this array. This result is the analog of the Dobinski formula Sum_{k >= 0} (k^n)*x^k/k! = exp(x)*Bell(n,x), where Bell(n,x) is the n-th row polynomial of A048993.
For example, for n = 6 we have S(6) = Sum_{k >= 1} k^6*(x^k/k!)^2 = (x^2 + 11*x^4 + x^6) * Sum_{k >= 0} (x^k/k!)^2 + (1/x)*(4*x^3 + 6*x^5) * Sum_{k >= 1} k*(x^k/k!)^2.
Setting x = 1 in the above result gives Sum_{k >= 0} k^n*/k!^2 = A000994(n)*Sum_{k >= 0} 1/k!^2 + A000995(n)*Sum_{k >= 1} k/k!^2. See A086880. (End)

			Triangle begins
n\k| 1    2    3    4    5    6    7    8
1  | 1
2  | 0    1
3  | 0    1    1
4  | 0    1    2    1
5  | 0    1    3    4    1
6  | 0    1    4   11    6    1
7  | 0    1    5   26   23    9    1
8  | 0    1    6   57   72   50   12    1
...
Connection constants: Row 6 = (0, 1, 4, 11, 6, 1) so
x^6 = x^2 + 4*x^2*(x - 1) + 11*x^2*(x - 1)^2 + 6*x^2*(x - 1)^2*(x - 2) + x^2*(x - 1)^2*(x - 2)^2.
Row 5 = [0, 1, 3, 4, 1]. There are 9 set partitions of {1,2,3,4,5} of the type described in the Name section:
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Number of      Set partitions                Count
blocks
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
2                {1}{2,3,4,5}                   1
3           {1}{2,4,5}{3}, {1}{2,3,5}{4},
            {1}{2,3,4}{5}                       3
4          {1}{2,3}{4}{5}, {1}{2,4}{3}{5},
           {1}{2,5}{3}{4}, {1}{2}{3}{4,5}       4
5          {1}{2}{3}{4}{5}                      1

Yue Cai and Margaret Readdy, Negative q-Stirling numbers, arXiv:1506.03249 [math.CO], 2015.
Emrah Kiliç and Helmut Prodinger, Identities with Squares of Binomial Coefficients: an Elementary and Explicit Approach, Publications de l'Institut Mathématique (Beograd) (N.S.), Vol.99(113) (2016), 243-248. See p. 248.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Tech Report TR 99-05, 1999.
E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.

Cf. A000295 (column 4), A007476 (row sums), A008277, A045618 (column 5), A048993, A246117 (unsigned matrix inverse), A256161, A000994, A000995, A086880.

Flatten[Table[Table[Sum[StirlingS2[j,Floor[k/2]] * StirlingS2[n-j-1,Floor[(k-1)/2]],{j,0,n-1}],{k,1,n}],{n,1,12}]] (* Vaclav Kotesovec, Feb 09 2015 *)

T(n,k) = Sum_{i = 0..n-1} Stirling2(i, floor(k/2))*Stirling2(n-i-1, floor((k - 1)/2)) for n,k >= 1.
Recurrence equation: T(1,1) = 1, T(n,1) = 0 for n >= 2; T(n,k) = 0 for k > n; otherwise T(n,k) = floor(k/2)*T(n-1,k) + T(n-1,k-1).
O.g.f. (with an extra 1): A(z) = 1 + Sum_{k >= 1} (x*z)^k/( ( Product_{i = 1..floor((k-1)/2)} (1 - i*z) ) * ( Product_{i = 1..floor(k/2)} (1 - i*z) ) ) = 1 + x*z + x^2*z^2 + (x^2 + x^3)*z^3 + (x^2 + 2*x^3 + x^4)*z^4 + .... satisfies A(z) = 1 + x*z + x^2*z^2/(1 - z)*A(z/(1 - z)).
k-th column generating function z^k/( ( Product_{i = 1..floor((k-1)/2)} (1 - i*z) ) * ( Product_{i = 1..floor(k/2)} (1 - i*z) ) ).
Recurrence for row polynomials: R(n,x) = x^2*Sum_{k = 0..n-2} binomial(n-2,k)*R(k,x) with initial conditions R(0,x) = 1 and R(1,x) = x. Compare with the recurrence satisfied by the Bell polynomials: Bell(n,x) = x*Sum_{k = 0..n-1} binomial(n-1,k) * Bell(k,x).
Row sums are A007476.

A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 6, 11, 23, 60, 179, 553, 1716, 5415, 17801, 61956, 228391, 882309, 3530322, 14531621, 61454091, 267479778, 1200680113, 5561767211, 26553471186, 130366882251, 656668581417, 3387887246292, 17886582294921, 96603394562849, 533645344137390, 3014295344076655

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..695

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346051, A346052.

nmax = 31; A[] = 0; Do[A[x] = x + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346050
    if (n<3): return (0,1,1)[n]
    else: return sum(binomial(n-3,k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 28 2022

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

A346051 G.f. A(x) satisfies: A(x) = 1 + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 5, 12, 28, 68, 181, 531, 1671, 5491, 18627, 65299, 237880, 903907, 3580619, 14729777, 62639952, 274442521, 1236730244, 5729809348, 27292248240, 133614280479, 671803041553, 3464970976743, 18309428363425, 99010800275743, 547462187824465, 3093329527120022

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..695

Cf. A000994, A000995, A000996, A000997, A000998.

Cf. A007476, A210540, A346050, A346052.

function a(n)
  if n lt 3 then return (1+(-1)^n)/2;
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022

nmax = 31; A[] = 0; Do[A[x] = 1 + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346051
    if (n<3): return (1, 0, 1)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

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

A346052 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 2, 3, 5, 11, 29, 80, 222, 630, 1881, 6004, 20420, 72979, 270659, 1035590, 4087205, 16675630, 70440641, 307933393, 1390117953, 6462787357, 30871458702, 151298796000, 760250325004, 3915477534861, 20662363081756, 111662169790416, 617482470676567, 3490973387652861

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..650

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346050, A346051.

function a(n) // a = A346052
  if n lt 3 then return Floor((3-n)/2);
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022

nmax = 31; A[] = 0; Do[A[x] = 1 + x + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[2] = 0; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346052
    if (n<3): return (1, 1, 0)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

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

A345177 a(0) = 1, a(1) = 0; a(n+2) = Sum_{k=0..n} Stirling2(n,k) * a(k).

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 8, 28, 149, 1029, 8039, 69375, 675541, 7584630, 98484836, 1457695370, 24117255106, 439505090491, 8756668806615, 190293641816660, 4508138040317573, 116298682305458460, 3258081214212853975, 98709283556190931672, 3219222306795403565116, 112538217720491999726102

Offset: 0

Ilya Gutkovskiy, Jun 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..355

Cf. A000994, A003659, A007469, A345178.

b:= proc(n, m) option remember; `if`(n=0,
      a(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> `if`(n<2, 1-n, b(n-2, 0)):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 13 2021

a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[StirlingS2[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = 1 + Normal[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!

cp's OEIS Frontend

A351151 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

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A351152 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

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

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

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A336970 G.f. A(x) satisfies: A(x) = 1 - x^2 * A(x/(1 - x)) / (1 - x).

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A246118 T(n,k), for n,k >= 1, is the number of partitions of the set [n] into k blocks, where, if the blocks are arranged in order of their minimal element, the odd-indexed blocks are all singletons.

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A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

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A346051 G.f. A(x) satisfies: A(x) = 1 + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

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A351151 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 5x)) / (1 - 5x).

A351152 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6x)) / (1 - 6x).