A001147 -id:A001147 - cp's OEIS frontend

A109714 Sequence defined by a recurrence close to that of A001147.

1, 1, 3, 18, 120, 1170, 12600, 176400, 2608200, 46607400, 883159200, 19429502400, 447567120000, 11629447830000, 316028116404000, 9516436753824000, 297478346845680000, 10151626256147376000, 359237701318479984000, 13733349319337487840000, 542212802070902202240000

Offset: 1

Niko Brummer (niko.brummer(AT)gmail.com), Aug 08 2005

From Christopher J. Smyth, Jan 26 2018: (Start)
The sequence is defined by the recurrence formula below. This recurrence is very similar to that of the sequence b(n) = A001147(n-1), which satisfies b(1)=1 and, for n >= 2, b(n) = Sum_{i=1..floor((n-1)/2)} binomial(n, i) * b(i) * b(n-i) + B, where B = 0 (n odd), = (1/2)*binomial(n, n/2)*b(n/2)^2 (n even) [see formula of Walsh on A001147 page]. Removal of the factor 1/2 from the definition of B gives, for n >= 3, the formula below for a(n).
This sequence seems to have been defined in the mistaken belief that it had applications. In fact the applications stated on earlier versions of this page actually belonged to A001147 -- see my comment on the A001147 page.
(End)

			a(3) = 3*a(1)*a(2) = 3, a(4) = 4*a(1)*a(3) + 6*a(2)^2 = 18.

Michael De Vlieger, Table of n, a(n) for n = 1..403

Cf. A001147

function m = a(n); if n==1 m = 1; elseif n==2 m = 1; else m = 0; for i=1:floor(n/2); f1 = binomial(n,i); f2 = a(i); f3 = a(n-i); m = m + f1*f2*f3; end; end;

Fold[Append[#1, Sum[Binomial[#2, i] #1[[i]] #1[[#2 - i]], {i, Floor[#2/2]}]] &, {1, 1}, Range[3, 21]] (* Michael De Vlieger, Dec 13 2017 *)

a(1) = 1, a(2) = 1 and a(n) = Sum_{i=1..floor(n/2)} binomial(n, i) * a(i) * a(n-i) for n >= 3.

A125080 Triangle, read by rows, defined by T(n,k) = A000108(n-k)A001147(k)C(n,2*k), for k=0..[n/2], n>=0, where A000108 is the Catalan numbers and A001147 is the double factorials.

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 14, 30, 6, 42, 140, 75, 132, 630, 630, 75, 429, 2772, 4410, 1470, 1430, 12012, 27720, 17640, 1470, 4862, 51480, 162162, 166320, 39690, 16796, 218790, 900900, 1351350, 623700, 39690, 58786, 923780, 4813380, 9909900, 7432425

Offset: 0

Paul D. Hanna, Nov 19 2006

			Table begins:
1;
1;
2, 1;
5, 6;
14, 30, 6;
42, 140, 75;
132, 630, 630, 75;
429, 2772, 4410, 1470;
1430, 12012, 27720, 17640, 1470;
4862, 51480, 162162, 166320, 39690;
16796, 218790, 900900, 1351350, 623700, 39690; ...

Cf. A115081 (row sums), A115080; A000108, A001147.

T(n,k)=binomial(2*n-2*k,n-k)/(n-k+1)*binomial(2*k,k)*k!/2^k*binomial(n,2*k)

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

Row sums equals A115081, which is column 0 of triangle A115080.

A165433 A transform of the double factorial numbers A001147.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 39, 97, 308, 897, 3139, 10304, 38997, 140893, 570002, 2230599, 9567979, 40091222, 181203603, 805962157, 3819522284, 17912075229, 88646095447, 435959031488, 2245454002137, 11530035000169, 61627679281154

Offset: 0

Paul Barry, Sep 18 2009

Hankel transform is A000178.

G. C. Greubel, Table of n, a(n) for n = 0..800

[(&+[Binomial(n-k,k)*Factorial(2*k)/(Factorial(k)*2^k): k in [0.. Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 20 2018

a:=proc(n) add(binomial(n-k,k)*factorial(2*k)/(factorial(k)*2^k),k=0..floor(n/2)) end proc: seq(a(n),n=0..30); # Muniru A Asiru, Oct 20 2018

Table[Sum[Binomial[n-k, k]*(2*k)!/(k!*2^k), {k, 0, Floor[n/2]}], {n,0, 30}] (* G. C. Greubel, Oct 20 2018 *)

vector(30, n, n--; sum(k=0, floor(n/2), binomial(n-k,k)*(2*k)!/(k!*2^k))) \\ G. C. Greubel, Oct 20 2018

G.f.: 1/(1-x-x^2-2x^4/(1-x-5x^2-12x^4/(1-x-9x^2-30x^4/(1-x-13x^2-56x^4/(1-.... (continued fraction);
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(2k)!/(k!*2^k).
Conjecture: 2*a(n) -3*a(n-1) +(3-2*n)*a(n-2) +(2*n-3)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 29 2013
a(n) ~ 2^(-1/2) * exp(sqrt(n)/2 - n/2 + 1/16) * n^(n/2) * (1 + 121/(192*sqrt(n))). - Vaclav Kotesovec, Apr 18 2024

A176231 Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147.

Original entry on oeis.org

1, -1, 1, 3, -6, 1, -15, 45, -15, 1, 105, -420, 210, -28, 1, -945, 4725, -3150, 630, -45, 1, 10395, -62370, 51975, -13860, 1485, -66, 1, -135135, 945945, -945945, 315315, -45045, 3003, -91, 1, 2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1

Offset: 0

Paul Barry, Apr 12 2010

Exponential Riordan array [1/sqrt(1+2x),x/(1+2x)]. Inverse of A176230.

Diagonal sums are an alternating sign version of A025164.

			Triangle begins
  1,
  -1, 1,
  3, -6, 1,
  -15, 45, -15, 1,
  105, -420, 210, -28, 1,
  -945, 4725, -3150, 630, -45, 1,
  10395, -62370, 51975, -13860, 1485, -66, 1,
  -135135, 945945, -945945, 315315, -45045, 3003, -91, 1,
  2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1
Production matrix is
  -1, 1,
  2, -5, 1,
  0, 12, -9, 1,
  0, 0, 30, -13, 1,
  0, 0, 0, 56, -17, 1,
  0, 0, 0, 0, 90, -21, 1,
  0, 0, 0, 0, 0, 132, -25, 1,
  0, 0, 0, 0, 0, 0, 182, -29, 1,
  0, 0, 0, 0, 0, 0, 0, 240, -33, 1

Cf. A001147, A025164, A176230.

T := (n,k) -> (2*n)!*(-1/2)^(n-k)/(2*k)!*(n-k)!:
seq(seq(T(n,k), k=0..n), n=0..8); # Peter Luschny, Jul 20 2019

(* The function RiordanArray is defined in A256893. *)
rows = 9;
R = RiordanArray[1/Sqrt[1 + 2 #]&, #/(1 + 2 #)&, rows, True];
R // Flatten (* Jean-François Alcover, Jul 20 2019 *)
T[ n_, k_] := Coefficient[ HermiteH[2 n, x/Sqrt[2]], x, 2 k]/2^n; (* Michael Somos, Jan 15 2020 *)
T[ n_, k_] := Coefficient[ Nest[# x - D[#, x]&, 1, 2 n], x, 2 k]; (* Michael Somos, Jan 15 2020 *)

{T(n, k) = my(t=1); for(i=1, 2*n, t = x*t - t'); polcoeff(t, 2*k)}; /* Michael Somos, Jan 15 2020 */

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

He_(2*n)(x) = Sum_{k=0..n} T(n, k)*x^(2*k) where He is Hermite's polynomial. - Michael Somos, Jan 15 2020

A273377 Discriminator of the double factorial of odd numbers (A001147).

Original entry on oeis.org

1, 3, 5, 11, 11, 11, 19, 23, 23, 41, 41, 64, 83, 89, 89, 89, 89, 89, 89, 89, 89, 109, 109, 167, 167, 167, 167, 167, 167, 167, 167, 167, 167, 167, 283, 283, 283, 283, 283, 283, 283, 283, 283, 283, 283, 379, 379, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421, 421

Offset: 1

Jeffrey Shallit, May 21 2016

nonn

The discriminator of a sequence is the least positive integer k such that the first n terms of the sequence are pairwise
incongruent, modulo k.
Here we are taking A001147 beginning at position 1.

A103512 Smallest m such that A001147(n)+m is prime.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 16, 8, 4, 64, 2, 2, 8, 2, 4, 118, 32, 82, 178, 4, 8, 4, 16, 16, 298, 64, 16, 194, 2, 298, 316, 8, 334, 32, 226, 386, 4, 2, 358, 8, 178, 254, 206, 206, 218, 8, 4, 254, 452, 914, 346, 2, 758, 394, 362, 394, 556, 422, 32, 346, 1108, 394, 932, 362, 604, 1382

Offset: 1

Lei Zhou, Feb 15 2005

nonn

			A001147(1)=1!!=1, 1+1=2 is prime, so a(1)=1;
A001147(2)=3!!=3, 3+2=5 is prime, so a(2)=2;
A001147(7)=13!!=135135, 135135+16=135151 is prime, so a(7)=16.

Cf. A103511, A001147.

Flatten[Table[Select[Range[1500],PrimeQ[(2n-1)!!+#]&,1],{n,66}]] (* James C. McMahon, Jan 20 2024 *)

A124495 G.f.: A(x) = 1/[1-x - Sum_{n>=1} A001147(n)x^(2n) ] where A001147(n) = (2n)!/(n!2^n) is the double factorials.

Original entry on oeis.org

1, 1, 2, 3, 8, 14, 43, 81, 283, 556, 2243, 4512, 21374, 43469, 243817, 497217, 3289606, 6697795, 51583952, 104698998, 922789643, 1867079621, 18522929815, 37380015420, 411572179999, 828925168492, 10014624164666, 20140445929353

Offset: 0

Paul D. Hanna, Nov 04 2006

nonn

Is this sequence equal to A076876 (meandric numbers for a river crossing two parallel roads at n points)?

			G.f.: A(x) = 1/(1-x - x^2 - 3*x^4 - 15*x^6 - 105*x^8 - 945*x^10 -...).

Cf. A001147.

a(n)=polcoeff(1/(1-x-sum(k=1,n\2,(2*k)!/k!/2^k*x^(2*k))+x*O(x^n)),n)

A143081 A symmetrical triangle of coefficients based on A001147: a(n)=(2n-1)a(n-1); t(n,m)=a(n)^2/((2n - 1)a(m)*a(n - m)).

Original entry on oeis.org

-1, 1, 1, 1, 3, 1, 3, 15, 15, 3, 15, 105, 175, 105, 15, 105, 945, 2205, 2205, 945, 105, 945, 10395, 31185, 43659, 31185, 10395, 945, 10395, 135135, 495495, 891891, 891891, 495495, 135135, 10395, 135135, 2027025, 8783775, 19324305, 24845535, 19324305, 8783775, 2027025, 135135, 2027025, 34459425

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 15 2008

Row sums are:{-1, 2, 5, 36, 415, 6510, 128709, 3065832, 85386015, 2721425850, 97665121125}.

			{-1},
{1, 1},
{1, 3, 1},
{3, 15, 15, 3},
{15, 105, 175, 105, 15},
{105, 945, 2205, 2205, 945, 105},
{945, 10395, 31185, 43659, 31185, 10395, 945},
{10395, 135135, 495495, 891891, 891891, 495495, 135135, 10395},
{135135, 2027025, 8783775, 19324305, 24845535, 19324305, 8783775, 2027025, 135135},
{2027025, 34459425, 172297125, 447972525, 703956825, 703956825, 447972525, 172297125, 34459425, 2027025}, {34459425, 654729075, 3710131425, 11130394275,
20670732225, 25264228275, 20670732225, 11130394275, 3710131425, 654729075,
34459425}

Cf. A001147.

a[0] = 1; a[n_] := a[n] = (2*n - 1)*a[n - 1]; Table[Table[a[n]^2/((2*n - 1)*a[m]*a[n - m]), {m, 0, n}], {n, 0, 10}]; Flatten[%]

a(n)=(2*n-1)*a(n-1); t(n,m)=a(n)^2/((2*n - 1)*a(m)*a(n - m)).

A144457 Coefficients of polynomials based on the generalized factorial at k=2 (A001147): b(n)=b(n-1+k; a(n)=b(n)a(n-1); p(x,n)=If[n == 0, 1, a(n - 1)(x - a(n - 1))*Product[x + 1/b(i), {i, 1, n - 1}]].

Original entry on oeis.org

1, -1, 1, -3, -8, 3, -15, -119, -217, 15, -105, -1574, -7440, -10954, 105, -945, -22679, -194646, -702874, -892281, 945, -10395, -363824, -4885615, -31288480, -94892945, -108046896, 10395, -135135, -6486479, -124999827, -1232430275, -6521470845, -17442096461, -18261339153, 135135

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 07 2008

The name contains an unmatched parenthesis. - Editors, Mar 13 2024
Row sums are:
{1, 0, -8, -336, -19968, -1812480, -239477760, -43588823040, -10461389783040, -3201186759966720, -1216451002230374400}.

			{1},
{-1, 1},
{-3, -8, 3},
{-15, -119, -217, 15},
{-105, -1574, -7440, -10954,105},
{-945, -22679, -194646, -702874, -892281,945},
{-10395, -363824, -4885615, -31288480, -94892945, -108046896, 10395},
{-135135, -6486479, -124999827, -1232430275, -6521470845, -17442096461, -18261339153, 135135}

Cf. A001147.

Clear[a, b, p, x, n]; k = 2; b[0] = 1; b[n_] := b[n] = b[n - 1] + k; a[0] = 1; a[n_] := a[n] = b[n]*a[n - 1]; p[x_, n_] = If[n == 0, 1, a[n - 1]*(x - a[n - 1])*Product[x + 1/b[i], {i, 1, n - 1}]]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]

b(n)=b(n-1+k; a(n)=b(n)*a(n-1); p(x,n)=If[n == 0, 1, a(n - 1)*(x - a(n - 1))*Product[x + 1/b(i), {i, 1, n - 1}]]; t(n,m)=coefficients(p(x,n)).

A208124 a(1)=2, a(n) = (4n/3)*(2n-1)!! (see A001147) for n>1.

Original entry on oeis.org

2, 8, 60, 560, 6300, 83160, 1261260, 21621600, 413513100, 8729721000, 201656555100, 5059746291600, 137034795397500, 3984550204635000, 123805667072587500, 4093840724533560000, 143540290403957947500, 5319434291440794525000, 207753461493493252837500, 8528826313943407221750000

Offset: 1

N. J. A. Sloane, Mar 27 2012

nonn

Bela Bollobas, The number of 1-factors in 2k-connected graphs, J. Combin. Theory Ser. B 25 (1978), no. 3, 363--366. MR0516268 (80m:05060). See Eq. (2).

a[n_] := If[n == 1, 2, (4n/3)*(2n - 1)!!]; Array[a, 20] (* Amiram Eldar, Dec 01 2018 *)

(n-1)*a(n) =n*(2n-1)*a(n-1), n>2. - R. J. Mathar, Mar 27 2012

cp's OEIS Frontend

A109714 Sequence defined by a recurrence close to that of A001147.

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A125080 Triangle, read by rows, defined by T(n,k) = A000108(n-k)*A001147(k)*C(n,2*k), for k=0..[n/2], n>=0, where A000108 is the Catalan numbers and A001147 is the double factorials.

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A165433 A transform of the double factorial numbers A001147.

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Magma

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A176231 Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147.

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A273377 Discriminator of the double factorial of odd numbers (A001147).

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A103512 Smallest m such that A001147(n)+m is prime.

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A124495 G.f.: A(x) = 1/[1-x - Sum_{n>=1} A001147(n)*x^(2n) ] where A001147(n) = (2n)!/(n!*2^n) is the double factorials.

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A143081 A symmetrical triangle of coefficients based on A001147: a(n)=(2*n-1)*a(n-1); t(n,m)=a(n)^2/((2*n - 1)*a(m)*a(n - m)).

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A144457 Coefficients of polynomials based on the generalized factorial at k=2 (A001147): b(n)=b(n-1+k; a(n)=b(n)*a(n-1); p(x,n)=If[n == 0, 1, a(n - 1)*(x - a(n - 1))*Product[x + 1/b(i), {i, 1, n - 1}]].

A125080 Triangle, read by rows, defined by T(n,k) = A000108(n-k)A001147(k)C(n,2*k), for k=0..[n/2], n>=0, where A000108 is the Catalan numbers and A001147 is the double factorials.

A124495 G.f.: A(x) = 1/[1-x - Sum_{n>=1} A001147(n)x^(2n) ] where A001147(n) = (2n)!/(n!2^n) is the double factorials.

A143081 A symmetrical triangle of coefficients based on A001147: a(n)=(2n-1)a(n-1); t(n,m)=a(n)^2/((2n - 1)a(m)*a(n - m)).

A144457 Coefficients of polynomials based on the generalized factorial at k=2 (A001147): b(n)=b(n-1+k; a(n)=b(n)a(n-1); p(x,n)=If[n == 0, 1, a(n - 1)(x - a(n - 1))*Product[x + 1/b(i), {i, 1, n - 1}]].