A082032 -id:A082032 - cp's OEIS frontend

A082021 a(0) = 7; for n > 0, a(n) is the greatest prime factor of PP(PP(a(n-1)))*a(n-1)+2 where PP(n) is an abbreviation for PreviousPrime(n).

7, 23, 131, 47, 643, 2459, 2000807, 503347241, 82125909539251, 9617692012399, 55555555342491359799151, 1116817987709786226917069, 578610396154837, 66992050984853, 254497141, 1660738053545999, 201525986561, 25600818891233, 796725607788661087, 23547857117470471

Offset: 0

Yasutoshi Kohmoto, May 10 2003

nonn

Tyler Busby, Table of n, a(n) for n = 0..25

Cf. A082032, A031441, A031442.

NestWhileList[FactorInteger[2+#*Prime[PrimePi[ # ]-2]][[ -1,1]] &, 7, True, 8] (* T. D. Noe, Nov 15 2006 *)
NestList[FactorInteger[NextPrime[NextPrime[#,-1],-1]#+2][[-1,1]]&,7,20] (* Harvey P. Dale, Dec 26 2017 *)

Description corrected by Rick L. Shepherd, Dec 19 2004
Corrected by T. D. Noe, Nov 15 2006
More terms from Harvey P. Dale, Dec 26 2017
a(19) from Tyler Busby, Oct 12 2023

A295100 a(n) = n! * [x^n] exp(nx)/(1 - 2x).

Original entry on oeis.org

1, 3, 20, 201, 2688, 44815, 894528, 20792205, 551518208, 16438822587, 543934387200, 19783668211153, 784536321392640, 33689132092480839, 1557397919735103488, 77117362592836807125, 4072280214605427376128, 228441851811771488284915, 13566762607790788699226112, 850372121882700252639269337

Offset: 0

Ilya Gutkovskiy, Nov 14 2017

nonn

The n-th term of the n-th binomial transform of A000165.

Robert Israel, Table of n, a(n) for n = 0..375
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Cf. A000165, A010844, A063170, A082032, A295098, A295099.

S:= series(exp(n*x)/(1-2*x),x,51):
seq(n!*coeff(S,x,n),n=0..50); # Robert Israel, Nov 14 2017

Table[n! SeriesCoefficient[Exp[n x]/(1 - 2 x), {x, 0, n}], {n, 0, 19}]

a(n) ~ 2^n * exp(n/2) * n!. - Vaclav Kotesovec, Nov 14 2017

a(n) = n! * Sum_{k=0..n} n^k*2^(n-k)/k! = 2^n*Gamma(n+1, n/2)*exp(n/2). - Robert Israel, Nov 14 2017

A136807 Hankel transform of double factorial numbers n!*2^n=A000165(n).

Original entry on oeis.org

1, 4, 256, 589824, 86973087744, 1282470362637926400, 2723154477021188283432960000, 1133321924829207204666583887642624000000, 120746421332702772771144114237731253721340313600000000

Offset: 0

Paul Barry, Jan 23 2008

By the properties of the Hankel transform, a(n)=2^(n(n+1))*A055209(n).

Also Hankel transform of A000354, A010844, A082032. - Philippe Deléham, Jan 23 2008

G. C. Greubel, Table of n, a(n) for n = 0..28
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000354, A010844, A082032, A055209.

[1] cat [(&*[(2*k)^(2*(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[(2k)^(2(n-k+1)),{k,n}],{n,0,10}] (* Harvey P. Dale, Apr 11 2013 *)

for(n=0,10, print1(prod(k=1,n,(2*k)^(2*(n-k+1))), ", ")) \\ G. C. Greubel, Oct 14 2018

a(n) = Product_{k=1..n} (2k)^(2(n-k+1)).

a(n) ~ 2^((n+1)^2) * Pi^(n+1) * n^(n^2 + 2*n + 5/6) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019

A097814 E.g.f. exp(3x)/(1-3x).

Original entry on oeis.org

1, 6, 45, 432, 5265, 79218, 1426653, 29961900, 719092161, 19415508030, 582465299949, 19221355075464, 691968783248145, 26986782548271978, 1133444867032206045, 51005019016463620932, 2448240912790296851457

Offset: 0

Paul Barry, Aug 26 2004

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A082032.

With[{nn=20},CoefficientList[Series[Exp[3x]/(1-3x),{x,0,nn}],x] Range[ 0,nn]!] (* or *) RecurrenceTable[{a[0]==1,a[n]==3n a[n-1]+3^n},a,{n,20}] (* Harvey P. Dale, Feb 23 2012 *)

a(n) = 3n*a(n-1)+3^n, n>0, a(0)=1; a(n) = 3^n*A000522(n).
G.f.: 1/Q(0), where Q(k) = 1 - 6*x*(k+1) - 9*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
Conjecture: a(n) +3*(-n-1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Dec 21 2014

A097815 E.g.f. exp(4x)/(1-4x).

Original entry on oeis.org

1, 8, 80, 1024, 16640, 333824, 8015872, 224460800, 7182811136, 258581463040, 10343259570176, 455103425282048, 21844964430315520, 1135938150443515904, 63612536425105326080, 3816752185507393306624, 244272139872477466591232

Offset: 0

Paul Barry, Aug 26 2004

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A082032, A097814.

With[{nn=20},CoefficientList[Series[Exp[4x]/(1-4x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 02 2017 *)

a(n) = 4n*a(n-1)+4^n, n>0, a(0)=1; a(n) = 4^n*A000522(n).
G.f.: 1/Q(0), where Q(k) = 1 - 8*x*(k+1) - 16*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
D-finite with recurrence a(n) +4*(-n-1)*a(n-1) +16*(n-1)*a(n-2)=0. - R. J. Mathar, Feb 19 2015

cp's OEIS Frontend

A082021 a(0) = 7; for n > 0, a(n) is the greatest prime factor of PP(PP(a(n-1)))*a(n-1)+2 where PP(n) is an abbreviation for PreviousPrime(n).

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A295100 a(n) = n! * [x^n] exp(n*x)/(1 - 2*x).

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Maple

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A136807 Hankel transform of double factorial numbers n!*2^n=A000165(n).

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Magma

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A097814 E.g.f. exp(3x)/(1-3x).

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A097815 E.g.f. exp(4x)/(1-4x).

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A295100 a(n) = n! * [x^n] exp(nx)/(1 - 2x).