A056040 -id:A056040 - cp's OEIS frontend

A163842 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial transform. Same as interpolating the beta numbers 1/beta(n,n) (A002457) with (A163869). Triangle read by rows, for n >= 0, k >= 0.

1, 7, 6, 43, 36, 30, 249, 206, 170, 140, 1395, 1146, 940, 770, 630, 7653, 6258, 5112, 4172, 3402, 2772, 41381, 33728, 27470, 22358, 18186, 14784, 12012, 221399, 180018, 146290, 118820, 96462, 78276, 63492, 51480

Offset: 0

Peter Luschny, Aug 06 2009

			Triangle begins:
      1;
      7,     6;
     43,    36,    30;
    249,   206,   170,   140;
   1395,  1146,   940,   770,   630;
   7653,  6258,  5112,  4172,  3402,  2772;
  41381, 33728, 27470, 22358, 18186, 14784, 12012;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Row sums are A163845. Cf. A056040, A163650, A163841, A163842, A163840, A002426, A000984.

# Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
a := n -> SumTria(k->swing(2*k+1),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n-k, n-i]*sf[2*i+1], {i, k, n}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).

A163076 Primes of the form k$ - 1. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

5, 19, 29, 139, 251, 12011, 48619, 51479, 155117519, 81676217699, 1378465288199, 5651707681619, 386971244197199, 1580132580471899, 30067266499541039, 6637553085023755473070799, 35257120210449712895193719, 399608854866744452032002440111

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			Since 4$ = 6 the prime 5 is listed.

Jinyuan Wang, Table of n, a(n) for n = 1..53
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A055490, A056040, A163078 (arguments k), A163074, A163075.

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

Reap[Do[f = n!/Quotient[n, 2]!^2; If[PrimeQ[p = f - 1], Sow[p]], {n, 1, 70}]][[2, 1]] // Union (* Jean-François Alcover, Jun 28 2013 *)

More terms from Jinyuan Wang, Mar 22 2020

A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

2, 3, 5, 31, 67, 139, 631, 9743, 16253, 17977, 27901, 37589

Offset: 1

Peter Luschny, Jul 21 2009

a(n) are the primes in A163077.

			5 is prime and 5$ + 1 = 30 + 1 = 31 is prime, so 5 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002981, A062363, A093804, A163077.

Cf. A056040. - Robert G. Wilson v, Aug 09 2010

a := proc(n) select(isprime,select(k -> isprime(A056040(k)+1),[$0..n])) end:

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; p = 2; lst = {}; While[p < 38000, a = f@p + 1; If[ PrimeQ@a, AppendTo[ lst, p]; Print@p]; p = NextPrime@p]; lst (* Robert G. Wilson v, Aug 08 2010 *)

is(k) = isprime(k) && ispseudoprime(1+k!/(k\2)!^2); \\ Jinyuan Wang, Mar 22 2020

a(8)-a(12) from Robert G. Wilson v, Aug 08 2010

A163213 Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 9, 13, 12, 2, 19, 2, 5, 36, 6, 19, 43, 11, 47, 67, 39, 41, 70, 12, 17, 83, 88, 81, 25, 53, 91, 97, 106, 79, 43, 39, 7, 29, 73, 6, 79, 115

Offset: 1

Peter Luschny, Jul 24 2009

nonn

If this is zero, p is a swinging Wilson prime.

			The swinging Wilson quotient related to the 5th prime is (252+1)/11=23, so the 5th term is 23 mod 11 = 1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A163211, A002068, A163210.

WR := proc(f,r,n) map(p->(f(p-1)+r(p))/p mod p,select(isprime,[$1..n])) end:
A002068 := n -> WR(factorial,p->1,n);
A163213 := n -> WR(swing,p->(-1)^iquo(p+2,2),n);

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; Mod[(sf[p - 1] + (-1)^Floor[(p + 2)/2])/p, p]); Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Jun 28 2013 *)

sf(n)=n!/(n\2)!^2
apply(p->sf(p-1)\/p%p, primes(100)) \\ Charles R Greathouse IV, Dec 11 2016

A163649 Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows.

Original entry on oeis.org

1, -1, 1, 1, -2, 2, -1, 3, -6, 6, 1, -4, 12, -24, 6, -1, 5, -20, 60, -30, 30, 1, -6, 30, -120, 90, -180, 20, -1, 7, -42, 210, -210, 630, -140, 140, 1, -8, 56, -336, 420, -1680, 560, -1120, 70

Offset: 0

Peter Luschny, Aug 02 2009

Given T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!, let A(n,k) = abs(T(n,k)) be the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*q^k. Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the extended Motzkin numbers A189912. (See A089627 for the Motzkin numbers and A194586 for the complementary Motzkin numbers.)

			1
-1, 1
1, -2, 2
-1, 3, -6, 6
1, -4, 12, -24, 6
-1, 5, -20, 60, -30, 30
1, -6, 30, -120, 90, -180, 20
-1, 7, -42, 210, -210, 630, -140, 140
1, -8, 56, -336, 420, -1680, 560, -1120, 70

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, The lost Catalan numbers.

Row sums give A163650, row sums of absolute values give A163865.

Aerated versions A194586 (odd case) and A089627 (even case).

a := proc(n,k) (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)! end:
seq(print(seq(a(n,k),k=0..n)),n=0..8);

t[n_, k_] := (-1)^(n - k)*Floor[k/2]!^(-2)*n!/(n - k)!; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

for(n=0,10, for(k=0,n, print1((-1)^(n -k)*( (floor(k/2))! )^(-2)*(n!/(n - k)!), ", "))) \\ G. C. Greubel, Aug 01 2017

T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!.

E.g.f.: egf(x,y) = exp(-x)*BesselI(0,2*x*y)*(1+x*y).

A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 6, -9, -7, -4, 0, 6, 44, 35, 28, 24, 24, 30, -165, -121, -86, -58, -34, -10, 20, 594, 429, 308, 222, 164, 130, 120, 140, -2037, -1443, -1014, -706, -484, -320, -190, -70, 70, 6824, 4787, 3344, 2330, 1624, 1140, 820, 630, 560, 630

Offset: 0

Peter Luschny, Aug 05 2009

An analog to the derangement triangle (A068106).

			1
0, 1
1, 1, 2
2, 3, 4, 6
-9, -7, -4, 0, 6
44, 35, 28, 24, 24, 30
-165, -121, -86, -58, -34, -10, 20

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Row sums are A163773.

Cf. A056040, A163650, A163771, A163772, A068106.

DiffTria := proc(f,n,display) local m,A,j,i,T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] - A[j] od;
for i from 0 to m do T := T,(-1)^(m-i)*A[i] od;
if display then print(seq(T[i],i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL,[T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
Computes n rows of the triangle.
A163770 := n -> DiffTria(k->swing(k),n,true);
A068106 := n -> DiffTria(k->factorial(k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).

A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040).

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 16, 11, 8, 6, 47, 31, 20, 12, 6, 146, 99, 68, 48, 36, 30, 447, 301, 202, 134, 86, 50, 20, 1380, 933, 632, 430, 296, 210, 160, 140, 4251, 2871, 1938, 1306, 876, 580, 370, 210, 70, 13102, 8851, 5980, 4042, 2736, 1860, 1280, 910, 700, 630

Offset: 0

Peter Luschny, Aug 06 2009

Triangle read by rows.

An analog to the binomial triangle of the factorials (A076571).

			Triangle begins
    1;
    2,   1;
    5,   3,   2;
   16,  11,   8,   6;
   47,  31,  20,  12,  6;
  146,  99,  68,  48, 36, 30;
  447, 301, 202, 134, 86, 50, 20;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Row sums are A163843.

Cf. A056040, A163865, A163841, A163842, A163650.

SumTria := proc(f,n,display) local m,A,j,i,T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] + A[j] od;
for i from 0 to m do T := T,A[i] od;
if display then print(seq(T[i],i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL,[T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
# Computes n rows of the triangle:
A163840 := n -> SumTria(swing,n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040), for n >= 0, k >= 0.

A163841 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984).

Original entry on oeis.org

1, 3, 2, 11, 8, 6, 45, 34, 26, 20, 195, 150, 116, 90, 70, 873, 678, 528, 412, 322, 252, 3989, 3116, 2438, 1910, 1498, 1176, 924, 18483, 14494, 11378, 8940, 7030, 5532, 4356, 3432, 86515, 68032, 53538

Offset: 0

Peter Luschny, Aug 06 2009

For n >= 0, k >= 0 let T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040). Triangle read by rows.

			Triangle begins
     1;
     3,    2;
    11,    8,    6;
    45,   34,   26,   20;
   195,  150,  116,   90,   70;
   873,  678,  528,  412,  322,  252;
  3989, 3116, 2438, 1910, 1498, 1176,  924;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Row sums are A163844. Cf. A056040, A163650, A163841, A163842, A163840, A026375, A002426, A000984.

Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
a := n -> SumTria(k->swing(2*k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

A180064 a(n) = n!/A056040(n).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 36, 36, 576, 576, 14400, 14400, 518400, 518400, 25401600, 25401600, 1625702400, 1625702400, 131681894400, 131681894400, 13168189440000, 13168189440000, 1593350922240000, 1593350922240000, 229442532802560000, 229442532802560000

Offset: 0

Robert G. Wilson v, Aug 08 2010

From Emeric Deutsch, Dec 24 2008 [edited and moved here by Andrey Zabolotskiy, Oct 19 2023]: (Start)
a(n+1) is the number of permutations of {1,2,...,n} with no even entry followed by a smaller entry. Example: a(5)=4 because we have 1234, 1324, 3124 and 2314.
a(n+1) is the number of permutations p of {1,2,...,n} such that p(j) is odd whenever j is even. Example: a(5)=4 because we have 4123, 2143, 2341 and 4321.
a(n+1) = A134434(n,0). (End)

Cf. A000142, A056040, A134434.

A180064 := n -> iquo(n,2)!^2; # Peter Luschny, Aug 23 2010

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k+1)), {k,n}]; Array[ #!/f@# &, 25, 0]

a(n) = A000142(n) / A056040(n).

a(n) = floor(n/2)!^2. - Peter Luschny, Aug 23 2010

A196747 Numbers n such that 3 does not divide swing(n) = A056040(n).

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 18, 19, 20, 24, 25, 26, 54, 55, 56, 60, 61, 62, 72, 73, 74, 78, 79, 80, 162, 163, 164, 168, 169, 170, 180, 181, 182, 186, 187, 188, 216, 217, 218, 222, 223, 224, 234, 235, 236, 240, 241, 242, 486, 487, 488, 492, 493, 494, 504, 505, 506, 510

Offset: 1

Peter Luschny, Oct 06 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Peter Luschny, On the prime factors of the swinging factorial.

Cf. A005836, A129508, A030979, A151750, A196748, A196749, A196750.

SwingExp := proc(m,n) local p, q; p := m;
do q := iquo(n,p);
   if (q mod 2) = 1 then RETURN(1) fi;
   if q = 0 then RETURN(0) fi;
   p := p * m;
od end:
Search := proc(n,L) local m, i, r; m := n;
for i in L do r := SwingExp(i,m);
   if r <> 0 then RETURN(NULL) fi
od; n end:
A196747_list := n -> Search(n,[3]):  # n is a search limit

(* A naive solution *) sf[n_] := n!/Quotient[n, 2]!^2; Select[Range[0, 600], ! Divisible[sf[#], 3] &] (* Jean-François Alcover, Jun 28 2013 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=my(t=valp(n,3)); t%2==0 && 2*valp(n\2,3)==t \\ Charles R Greathouse IV, Feb 02 2016

cp's OEIS Frontend

A163842 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial transform. Same as interpolating the beta numbers 1/beta(n,n) (A002457) with (A163869). Triangle read by rows, for n >= 0, k >= 0.

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A163076 Primes of the form k$ - 1. Here '$' denotes the swinging factorial function (A056040).

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A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

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A163213 Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function (A056040).

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A163649 Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows.

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A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).

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A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040).

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