A001516 -id:A001516 - cp's OEIS frontend

A144659 a(n) = A001516(n)/2.

0, 0, 3, 60, 990, 16485, 292215, 5602338, 116549370, 2627202105, 63960690420, 1675359272730, 47031228602358, 1409683851264780, 44956320071089245, 1520493296271210030, 54376042036599780570, 2050556012681642525763, 81336729496634645414265, 3385729099078381669910310

Offset: 0

N. J. A. Sloane, Jan 30 2009

nonn

Also a(n) = 3*A144507(n).

Also a(n) = Sum_{k} binomial(k-n,2) T(n,k) where T(n,k) is the triangle in A144331.

A001514 Bessel polynomial {y_n}'(1).

Original entry on oeis.org

0, 1, 9, 81, 835, 9990, 137466, 2148139, 37662381, 733015845, 15693217705, 366695853876, 9289111077324, 253623142901401, 7425873460633005, 232122372003909045, 7715943399320562331, 271796943164015920914, 10114041937573463433966

Offset: 0

N. J. A. Sloane

nonn

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A001516, A001518, A065920, A144505.

(As in A001497 define:) f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
[seq( subs(x=1,diff(f(n),x)),n=0..60)];
f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..60)]; # uses a different offset

Table[Sum[(n+k+1)!/((n-k-1)!*k!*2^(k+1)), {k,0,n-1}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0}, Table[n*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[1 - n, -2*n, 2], {n,1,50}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,50, print1(sum(k=0,n-1, (n+k+1)!/((n-k-1)!*k!*2^(k+1))), ", ")) \\ G. C. Greubel, Aug 14 2017

a(n) = (1/2) * Sum_{k=0..n} (n+k+2)!/((n-k)!*k!*2^k) (with a different offset).
D-finite with recurrence: (n-1)^2 * a(n) = (2*n-1)*(n^2 - n + 1)*a(n-1) + n^2*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ 2^(n+1/2) * n^(n+1) / exp(n-1). - Vaclav Kotesovec, Jul 22 2015
a(n) = n*2^n*(1/2){n}*hypergeometric1f1(1-n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 14 2017
From G. C. Greubel, Aug 16 2017: (Start)
G.f.: (1/(1-t))*hypergeometric2f0(2, 3/2; -; 2*t/(1-t)^2).
E.g.f.: (1 - 2*x)^(-3/2)*((1 - x)*sqrt(1 - 2*x) + (3*x - 1))*exp((1 - sqrt(1 - 2*x))). (End)

A144507 Column 4 of triangle in A144505.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 20, 330, 5495, 97405, 1867446, 38849790, 875734035, 21320230140, 558453090910, 15677076200786, 469894617088260, 14985440023696415, 506831098757070010, 18125347345533260190, 683518670893880841921, 27112243165544881804755, 1128576366359460556636770

Offset: 0

N. J. A. Sloane, Dec 14 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A001516, A144505, A144506.

I:=[0,0,0,0,0,1]; [n le 6 select I[n] else ((2*n-9)*(n^2-9*n+22)*Self(n-1) + (n-3)*(n-4)*Self(n-2))/((n-5)*(n-6)): n in [1..32]]; // G. C. Greubel, Oct 10 2023

f4:=proc(n) local k; add((n+k-1)!/(4!*(n-k-5)!*k!*2^k),k=0..n-5); end;
[seq(f4(n), n=0..60)];

Table[Sum[1/6 (n+k+2)!/(2^(k+2) (n-k-2)! k!), {k,0,n-2}], {n, -3, 20}] (* Vincenzo Librandi, Jan 27 2020 *)

@CachedFunction
def A144507(n): return sum(binomial(n-5,j)*rising_factorial(n-4,j+4)/(24*2^j) for j in range(n-4))
[A144507(n) for n in range(31)] # G. C. Greubel, Oct 10 2023

a(n) = (1/4!)*Sum_{k=0..n-5} (n+k-1)!/((n-k-5)!*k!*2^k).
a(n) = A001516(n-3)/6 for n > 2. [Corrected by Georg Fischer, Jan 25 2020]
a(n) = ( (2*n-7)*(n^2 -7*n +14)*a(n-1) + (n-2)*(n-3)*a(n-2) )/((n-4)*(n-5)), with a(0)=a(1)=a(2)=a(3)=a(4)=0, and a(5)=1. - G. C. Greubel, Oct 10 2023

A065944 Bessel polynomial {y_n}''(-1).

Original entry on oeis.org

0, 0, 6, -60, 720, -9870, 153510, -2679264, 51934680, -1107917910, 25807660560, -651977992380, 17758547202396, -518856566089680, 16188283372489410, -537210169663283760, 18894951642157260480, -702160022681408982114

Offset: 0

N. J. A. Sloane, Dec 08 2001

sign

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Cf. A001518, A001516.

f:=Factorial;; Concatenation([0,0], List([2..20], n-> Sum([0..n-2], k-> (-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2)*f(k)) ))); # G. C. Greubel, Jul 10 2019

f:=Factorial; [0,0] cat [(&+[((-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2) *f(k))): k in [0..n-2]]): n in [2..20]]; // G. C. Greubel, Jul 10 2019

Table[Sum[(n+k+2)!*(-1)^k/(2^(k+2)*(n-k-2)!*k!), {k,0,n-2}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0, 0}, Table[4*n*(n-1)*Pochhammer[1/2, n]*(-2)^(n-2)* Hypergeometric1F1[2-n, -2*n, -2], {n, 2,20}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,20, print1(sum(k=0,n-2, (n+k+2)!*(-1)^k/(2^(k+2)*(n-k-2)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017

f=factorial; [0,0]+[sum((-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2)*f(k)) for k in (0..n-2)) for n in (2..20)] # G. C. Greubel, Jul 10 2019

Recurrence: (n-2)*(n-1)*a(n) = -(n-2)*(n+1)*(2*n-1)*a(n-1) + n*(n+1)*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ (-1)^n * 2^(n+1/2) * n^(n+2) / exp(n+1). - Vaclav Kotesovec, Jul 22 2015
From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 2*n*(n-1)*(1/2){n}*(-2)^(n - 1)* hypergeometric1f1(2 - n, -2*n, -2), where (a){n} is the Pochhammer symbol.
E.g.f.: (1 + 2*x)^(-5/2)*(x*(x + 2)*sqrt(1 + 2*x) + (2*x^3 - 2*x)) * exp(-1 + sqrt(1 + 2*x)). (End)
G.f.: (6*x^2/(1-x)^5)*hypergeometric2f0(3,5/2; - ; -2*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A065945 Bessel polynomial {y_n}''(2).

Original entry on oeis.org

0, 0, 6, 210, 6390, 201810, 6895140, 257335596, 10489055220, 465303486780, 22363517407770, 1159112646836430, 64499453473280826, 3837361123234687230, 243168894263042103720, 16356164256377393353080, 1164094991704907423494920

Offset: 0

N. J. A. Sloane, Dec 08 2001

nonn

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Cf. A001518, A001516.

Join[{0, 0}, Table[4*n*(n - 1)*Pochhammer[1/2, n]*4^(n - 2)* Hypergeometric1F1[2 - n, -2*n, 1], {n,2,50}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))), ", ")) \\ G. C. Greubel, Aug 14 2017

From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 4*n*(n - 1)*(1/2)_{n}*4^(n - 2)*hypergeometric1f1(2-n, -2*n, 1).
E.g.f.: (-1/16)*(1 - 4*x)^(-5/2)*((56*x^2 - 44*x + 6)*sqrt(1 - 4*x) + (16*x^3 - 180*x^2 + 56*x - 6))*exp((1 - sqrt(1 - 4*x))/2). (End)
G.f.: (6*x^2/(1-x)^5)*hypergeometric2f0(3,5/2; - ; 4*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A065946 Bessel polynomial {y_n}''(-2).

Original entry on oeis.org

0, 0, 6, -150, 3870, -110670, 3538500, -125941284, 4953759300, -213744815460, 10047637214010, -511403305348650, 28029852267603186, -1646397200571955650, 103190849406195456360, -6875135229835376875560, 485256294032090950981800

Offset: 0

N. J. A. Sloane, Dec 08 2001

sign

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Cf. A001518, A001516.

Join[{0, 0}, Table[4*n*(n - 1)*Pochhammer[1/2, n]*(-4)^(n - 2)*
Hypergeometric1F1[2 - n, -2*n, -1], {n,2,50}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))*(-1)^k), ", ")) \\ G. C. Greubel, Aug 14 2017

From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 4*n*(n - 1)*(1/2){n}*(-4)^(n - 2)*hypergeometric1f1(2-n, -2*n, -1), where (a){n} is the Pochhammer symbol.
E.g.f.: (1/16)*(1 + 4*x)^(-5/2)*((24*x^2 + 20*x + 2)*sqrt(1 + 4*x) + (16*x^3 - 12*x^2 - 24*x - 2))*exp((sqrt(1 + 4*x) -1)/2). (End)
G.f.: (6*x^2/(1-x)^5)*hypergeometric2f0(3,5/2; - ; -4*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A065947 Bessel polynomial {y_n}''(3).

Original entry on oeis.org

0, 0, 6, 300, 13320, 620130, 31406550, 1743174216, 105889417200, 7010411889690, 503353562247360, 39003404559533700, 3246506259033473436, 289042023964190515200, 27418894569798460848210, 2761554229456140638184840, 294364593823858690215256200

Offset: 0

N. J. A. Sloane, Dec 08 2001

nonn

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Cf. A001516, A001518.

Join[{0, 0}, Table[4*n*(n - 1)*Pochhammer[1/2, n]*6^(n - 2)* Hypergeometric1F1[2 - n, -2*n, 2/3], {n, 2, 50}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))*(3/2)^k), ", ")) \\ G. C. Greubel, Aug 14 2017

From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 4*n*(n - 1)*(1/2){n}*6^(n - 2)*hypergeometric1F1(2-n, -2*n, 2/3), where (a){n} is the Pochhammer symbol.
E.g.f.: (-1/81)*(1 - 6*x)^(-5/2)*((171*x^2 - 90*x + 8)*sqrt(1 - 6*x) + (54*x^3 - 648*x^2 + 114*x - 8))*exp((1 - sqrt(1 - 6*x))/3). (End)
G.f.: (6*x^2/(1-x)^5)*hypergeometric2F0(3,5/2; - ; 6*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A065948 Bessel polynomial {y_n}''(-3).

Original entry on oeis.org

0, 0, 6, -240, 9540, -415590, 20134590, -1082674404, 64221641820, -4173853100670, 295282282905720, -22605059036265420, 1862664627479732076, -164425432052147568120, 15483794266369962976170, -1549617160894627918342620, 164264715996348003982855020

Offset: 0

N. J. A. Sloane, Dec 08 2001

sign

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Cf. A001518, A001516.

Join[{0, 0}, Table[4*n*(n - 1)*Pochhammer[1/2, n]*(-6)^(n - 2)* Hypergeometric1F1[2 - n, -2*n, -2/3], {n, 2, 50}]] (* G. C. Greubel, Aug 15 2017 *)

for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))*(-3/2)^k ), ", ")) \\ G. C. Greubel, Aug 15 2017

From G. C. Greubel, Aug 15 2017: (Start)
a(n) = 4*n*(n - 1)*(1/2){n}*(-6)^(n - 2)* hypergeometric1f1(2 - n; -2*n; -2/3), where (a){n} is the Pochhammer symbol.
E.g.f.: (-1/81)*(1 + 6*x)^(-5/2)*((-99*x^2 - 54*x - 4)*sqrt(1 + 6*x) + (-54*x^3 + 66*x + 4))*exp(-(1 - sqrt(1 + 6*x))/3). (End)
G.f.: (6*x^2/(1-x)^5)*hypergeometric2f0(3,5/2; - ; -6*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A065949 Bessel polynomial {y_n}'''(0).

Original entry on oeis.org

0, 0, 0, 90, 630, 2520, 7560, 18900, 41580, 83160, 154440, 270270, 450450, 720720, 1113840, 1670760, 2441880, 3488400, 4883760, 6715170, 9085230, 12113640, 15939000, 20720700, 26640900, 33906600, 42751800, 53439750, 66265290, 81557280, 99681120, 121041360, 146084400

Offset: 0

N. J. A. Sloane, Dec 08 2001

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Bessel functions or polynomials
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001518, A001516.

Drop[90*Binomial[Range[40]-3,6],5] (* Harvey P. Dale, Sep 20 2013 *)

for(n=0,50, print1(90*binomial(n+3,6), ", ")) \\ G. C. Greubel, Aug 15 2017

a(n) = 90 * C(n-3, 6) = 90 * A000579(n-3). - Ralf Stephan, Sep 03 2003
From Colin Barker, Aug 01 2013: (Start)
a(n) = ((-2+n)*(-1+n)*n*(1+n)*(2+n)*(3+n))/8.
G.f.: -90*x^3 / (x-1)^7. (End)
E.g.f.: (1/8)*x^3*(120 + 90*x + 18*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 15 2017

More terms from Colin Barker, Aug 01 2013

A065950 Bessel polynomial {y_n}'''(1).

Original entry on oeis.org

0, 0, 0, 90, 3150, 81900, 1992060, 48771450, 1237774230, 32978969100, 927339227100, 27566149731120, 866148362679600, 28735959507074820, 1005105838958594100, 36999204981675832350, 1430792213377354462530, 58019598569681129648700

Offset: 0

N. J. A. Sloane, Dec 08 2001

nonn

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Cf. A001516, A001518.

[0,0,0] cat [(&+[Binomial(n-3,k)*Factorial(n+k+3)/(2^(k+3) * Factorial(n-3)): k in [0..n-3]]): n in [3..30]]; // G. C. Greubel, Sep 23 2023

Join[{0,0,0}, Table[6*Binomial[n,3]*Pochhammer[1/2,n]*2^n* Hypergeometric1F1[3-n,-2*n,2], {n,3,50}]] (* G. C. Greubel, Aug 15 2017 *)
CoefficientList[Series[(90*t^3/(1-t)^7)*HypergeometricPFQ[{4, 7/2}, {}, 2*t/(1-t)^2], {t,0,50}], t] (* G. C. Greubel, Aug 16 2017 *)

for(n=0,50, print1(sum(k=0,n-3, ((n+k+3)!/(2^(k+3)*k!*(n-k-3)!))), ", ")) \\ G. C. Greubel, Aug 15 2017

def A065950(n): return sum(binomial(n-3,k)*rising_factorial(n-2,k+6)//2^(k+3) for k in range(n-2))
[A065950(n) for n in range(31)] # G. C. Greubel, Sep 23 2023

a(n) = 6*binomial(n, 3)*(1/2){n}*2^n*hypergeometric1f1(3-n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 15 2017
G.f.: (90*x^3/(1-x)^7)*hypergeometric2f0(4,7/2; - ; 2*x/(1-x)^2). - G. C. Greubel, Aug 16 2017
a(n) ~ 2^(n + 1/2) * n^(n+3) / exp(n-1). - Vaclav Kotesovec, Jun 09 2019

cp's OEIS Frontend

A144659 a(n) = A001516(n)/2.

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A001514 Bessel polynomial {y_n}'(1).

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A144507 Column 4 of triangle in A144505.

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A065944 Bessel polynomial {y_n}''(-1).

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A065945 Bessel polynomial {y_n}''(2).

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A065946 Bessel polynomial {y_n}''(-2).

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A065947 Bessel polynomial {y_n}''(3).

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A065948 Bessel polynomial {y_n}''(-3).

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