cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A163087 Product{k|n} k$. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 240, 140, 840, 3780, 15120, 2772, 221760, 12012, 960960, 9266400, 10810800, 218790, 7351344000, 923780, 16761064320, 3259095840, 3910915008, 16224936, 41977154419200, 2028117000, 249864014400
Offset: 0

Views

Author

Peter Luschny, Jul 21 2009

Keywords

Examples

			The set of positive divisors of 3 is {1,3}. Thus a(3) = 1$ * 3$ = 1 * 6 = 6.

Links

Crossrefs

Cf. A098710, A163088, A163089.

Programs

  • Maple
    a := proc(n) local i; mul(i,i=map(swing,numtheory[divisors](n))) end:
  • Mathematica
    sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Product[sf[k], {k, Divisors[n]}]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 26 2013 *)

A163088 (Product{k|n} k$) / n$. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 12, 1, 12, 6, 60, 1, 240, 1, 280, 180, 840, 1, 151200, 1, 90720, 840, 5544, 1, 15523200, 30, 24024, 3780, 5765760, 1, 93405312000, 1, 10810800, 16632, 437580, 4200, 6792641856000, 1, 1847560, 72072, 1173274502400
Offset: 0

Views

Author

Peter Luschny, Jul 21 2009

Keywords

Links

Crossrefs

Cf. A163087, A163089.

Programs

  • Maple
    a := n -> A163087(n) / A056040(n):
  • Mathematica
    sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Product[sf[k], {k, Divisors[n]}]/sf[n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 26 2013 *)

A163089 n! / (Product{k|n} k$). Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 3, 36, 48, 96, 240, 14400, 2160, 518400, 90720, 141120, 1935360, 1625702400, 870912, 131681894400, 145152000, 15676416000, 287400960000, 1593350922240000, 14780620800, 7648084426752000, 1614043791360000
Offset: 0

Views

Author

Peter Luschny, Jul 21 2009

Keywords

Links

Crossrefs

Cf. A163087, A163088.

Programs

  • Maple
    a := n -> n! / A163087(n):
  • Mathematica
    sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := n!/Product[sf[k], {k, Divisors[n]}]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 26 2013 *)

A194586 Triangle read by rows, T(n,k) the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*(k mod 2)*q^k.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 3, 0, 6, 0, 4, 0, 24, 0, 0, 5, 0, 60, 0, 30, 0, 6, 0, 120, 0, 180, 0, 0, 7, 0, 210, 0, 630, 0, 140, 0, 8, 0, 336, 0, 1680, 0, 1120, 0, 0, 9, 0, 504, 0, 3780, 0, 5040, 0, 630, 0, 10, 0, 720, 0, 7560, 0, 16800, 0, 6300, 0, 0, 11, 0, 990, 0, 13860, 0, 46200, 0, 34650, 0, 2772, 0, 12
Offset: 0

Views

Author

Peter Luschny, Aug 29 2011

Keywords

Comments

Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the complementary Motzkin numbers A005717. (See A089627 for the Motzkin numbers and A163649 for the extended Motzkin numbers.)

Examples

			               0
              0, 1
            0, 2, 0
           0, 3, 0, 6
         0, 4, 0, 24, 0
       0, 5, 0, 60, 0, 30
    0, 6, 0, 120, 0, 180, 0
  0, 7, 0, 210, 0, 630, 0, 140
                0
                q
               2 q
            3 q + 6 q^3
           4 q + 24 q^3
       5 q + 60 q^3  + 30 q^5
      6 q + 120 q^3  + 180 q^5
  7 q + 210 q^3  + 630 q^5  + 140 q^7

Links

Crossrefs

Row sums are A109188. Cf. A056040, A005717, A163649, A089627.

Programs

  • Maple
    A194586 := proc(n,k) local j, swing; swing := n -> n!/iquo(n,2)!^2:
add(binomial(n,j)*swing(j)*q^j*(j mod 2),j=0..n); coeff(%,q,k) end:
seq(print(seq(A194586(n,k),k=0..n)),n=0..8);
  • Mathematica
    sf[n_] := n!/Quotient[n, 2]!^2;
row[n_] := Sum[Binomial[n, j] sf[j] q^j Mod[j, 2], {j, 0, n}] // CoefficientList[#, q]& // PadRight[#, n+1]&;
Table[row[n], {n, 0, 12}] (* Jean-François Alcover, Jun 26 2019 *)

Formula

egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).

A194590 a(n) = (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2.

Original entry on oeis.org

0, 0, 1, -2, 7, -14, 38, -76, 187, -374, 874, -1748, 3958, -7916, 17548, -35096, 76627, -153254, 330818, -661636, 1415650, -2831300, 6015316, -12030632, 25413342, -50826684, 106853668, -213707336, 447472972, -894945944, 1867450648, -3734901296, 7770342787
Offset: 0

Views

Author

Peter Luschny, Aug 30 2011

Keywords

Comments

The binomial transform of a(n) are the complementary Riordan numbers A194589 (see link).

Links

Crossrefs

Cf. A107373 (has offset 1).

Programs

  • Maple
    A056040 := n -> n!/iquo(n,2)!^2:
A152271 := n -> `if`(n mod 2 = 0, 1, (n+1)/2):
A194590 := n -> (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2:
  • Mathematica
    sf[n_] := n!/Quotient[n, 2]!^2;
a[n_] := (-1)^n (sf[n + 1] * If[EvenQ[n], 1, (n + 1)/2] - 2^n)/2;
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 26 2019 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*cr(k), where cr(k) are the complementary Riordan numbers A194589.

A195009 Triangle read by rows, T(n,k) = k^n*A056040(n), n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 8, 0, 6, 48, 162, 0, 6, 96, 486, 1536, 0, 30, 960, 7290, 30720, 93750, 0, 20, 1280, 14580, 81920, 312500, 933120, 0, 140, 17920, 306180, 2293760, 10937500, 39191040, 115296020, 0, 70, 17920, 459270, 4587520, 27343750, 117573120, 403536070, 1174405120
Offset: 0

Views

Author

Peter Luschny, Sep 07 2011

Keywords

Examples

			                     1
                    0, 1
                  0, 2, 8
               0, 6, 48, 162
            0, 6, 96, 486, 1536
       0, 30, 960, 7290, 30720, 93750
0, 20, 1280, 14580, 81920, 312500, 933120

Links

Programs

  • Maple
    swing := n -> n!/iquo(n,2)!^2: pow := (n,k) -> if k=0 and n=0 then 1 else n^k fi: A195009 := (n,k) -> pow(k,n)*swing(n):
# Formula:
omega := proc(x) BesselI(0,2*m*x)+(2*m*x+1)*BesselI(1,2*m*x) end:
f := n -> `if`(irem(n,2)=1,(n+1)/2,1/(n+1)): A195009 := proc(n,k)
limit(f(n)*(D@@n)(omega)(x),x=0); subs(m=k,%) end;
  • Mathematica
    sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; t[0, 0] = 1; t[n_, k_] := k^n*sf[n]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Formula

T(n,k) = f(n)*lim(x=0, (d^n/dx)(BesselI(0,2*k*x)+(2*k*x+1) *BesselI(1,2*k*x) where f(n) = (n+1)/2 if n is odd, 1/(n+1) otherwise.

A321625 The Riordan square of the swinging factorial (A056040), triangle read by rows, T(n, k) for 0 <= k<= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 10, 5, 1, 6, 22, 22, 7, 1, 30, 66, 66, 38, 9, 1, 20, 140, 218, 146, 58, 11, 1, 140, 372, 574, 542, 270, 82, 13, 1, 70, 826, 1680, 1708, 1134, 446, 110, 15, 1, 630, 1930, 4156, 5432, 4126, 2106, 682, 142, 17, 1
Offset: 0

Views

Author

Peter Luschny, Nov 22 2018

Keywords

Examples

			[0] [   1]
[1] [   1,    1]
[2] [   2,    3,    1]
[3] [   6,   10,    5,    1]
[4] [   6,   22,   22,    7,    1]
[5] [  30,   66,   66,   38,    9,    1]
[6] [  20,  140,  218,  146,   58,   11,    1]
[7] [ 140,  372,  574,  542,  270,   82,   13,   1]
[8] [  70,  826, 1680, 1708, 1134,  446,  110,  15,  1]
[9] [ 630, 1930, 4156, 5432, 4126, 2106,  682, 142, 17,  1]

Crossrefs

T(n, 0) = A056040 (swinging factorial), A321626 (row sums), A000007 (alternating row sums).
Cf. A321620.

Programs

  • Maple
    # The function RiordanSquare is defined in A321620.
SwingingFactorial := (1 + x/(1 - 4*x^2))/sqrt(1 - 4*x^2);
RiordanSquare(SwingingFactorial, 10);
  • Mathematica
    (* The function RiordanSquare is defined in A321620. *)
SwingingFactorial = (1 + x/(1 - 4*x^2))/Sqrt[1 - 4*x^2];
RiordanSquare[SwingingFactorial, 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)
  • Sage
    # uses[riordan_square from A321620]
riordan_square((1 + x/(1 - 4*x^2))/sqrt(1 - 4*x^2), 10)

A350464 Table read by rows. Interpolating the swinging factorial (A056040) and the double factorial (A001147).

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 2, 15, 15, 0, 6, 91, 210, 105, 0, 6, 690, 2835, 3150, 945, 0, 30, 5214, 42405, 79695, 51975, 10395, 0, 20, 44772, 666666, 2057055, 2207205, 945945, 135135, 0, 140, 384756, 11274900, 54879825, 90090000, 62432370, 18918900, 2027025
Offset: 0

Views

Author

Peter Luschny, Mar 13 2022

Keywords

Examples

			Triangle starts:
[0] 1;
[1] 0,  1;
[2] 0,  1,   3;
[3] 0,  2,   15,     15;
[4] 0,  6,   91,     210,     105;
[5] 0,  6,   690,    2835,    3150,     945;
[6] 0,  30,  5214,   42405,   79695,    51975,    10395;
[7] 0,  20,  44772,  666666,  2057055,  2207205,  945945,  135135;

Crossrefs

Cf. A350465 (row sums), A350466 (alternating row sums).
Cf. A056040, A001147, A001880.

Programs

  • Mathematica
    Swing[n_] := n! / Floor[n/2]!^2;
Z[n_] := Flatten[Table[{0, Swing[j]}, {j, 0, n}]];
T[n_, k_] := BellY[2 n, k, Z[n - k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

Formula

The partial Bell polynomials Y_{2*n, k}(Z) applied to the list Z of the aerated swinging factorials (A056040).

A163081 Primes of the form p$ + 1 where p is prime, where '$' denotes the swinging factorial (A056040).

Original entry on oeis.org

3, 7, 31, 4808643121, 483701705079089804581, 3283733939424401442167506310317720418331001
Offset: 1

Views

Author

Peter Luschny, Jul 21 2009

Keywords

Comments

The values of p are 2, 3, 5, 31, 67, 139 which is A163079. Subsequence of A163075 (primes of the form k$ + 1).

Examples

			3 and 3$ + 1 = 7 are prime, so 7 is a member.

Links

Crossrefs

Cf. A056040, A103319, A163074-A163083.

Programs

  • Maple
    a := proc(n) select(isprime,[$2..n]); select(isprime, map(x -> A056040(x)+1,%)) end:

A163082 Primes of the form p$ - 1 where p is prime, where '$' denotes the swinging factorial (A056040).

Original entry on oeis.org

5, 29, 139, 12011, 5651707681619, 386971244197199, 35257120210449712895193719, 815027488562171580969632861193966578650499
Offset: 1

Views

Author

Peter Luschny, Jul 21 2009

Keywords

Comments

The first values of p are 3, 5, 7, 13, 41 from A163080. Subsequence of A163076 (primes of the form k$ - 1).

Examples

			3 and 3$ - 1 = 5 are prime, so 5 is a member.

Links

Crossrefs

Cf. A056040, A163074-A163083.

Programs

  • Maple
    a := proc(n) select(isprime,[$2..n]); select(isprime, map(x -> A056040(x)-1,%)) end:
Previous Showing 31-40 of 146 results. Next

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