login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A152818 Array read by antidiagonals: T(n,k) = (k+1)^n*(n+k)!/n!. 13
1, 1, 1, 1, 4, 2, 1, 12, 18, 6, 1, 32, 108, 96, 24, 1, 80, 540, 960, 600, 120, 1, 192, 2430, 7680, 9000, 4320, 720, 1, 448, 10206, 53760, 105000, 90720, 35280, 5040, 1, 1024, 40824, 344064, 1050000, 1451520, 987840, 322560, 40320 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A009998/A119502 gives triangle of unreduced coefficients of polynomials defined by A152650/A152656. a(n) gives numerators with denominators n! for each row.

Row 0 is A000142. Row 1 is formed from positive members of A001563. Row 2 is A055533. Column 0 is A000012. Column 1 is formed from positive members of A001787. Column 2 is A006043. Column 3 is A006044. - Omar E. Pol, Jan 06 2009

LINKS

Table of n, a(n) for n=0..44.

F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. See page 422.

F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)

F. A. Haight, Letter to N. J. A. Sloane, n.d.

FORMULA

E.g.f. for array as a triangle: exp(x)/(1-t*x*exp(x)) = 1+(1+t)*x+(1+4*t+2*t^2)*x^2/2!+(1+12*t+18*t^2+6*t^3)*x^3/3!+.... E.g.f. is int {z = 0..inf} exp(-z)*F(x,t*z), (x and t chosen sufficiently small for the integral to converge), where F(x,t) = exp(x*(1+t*exp(x))) is the e.g.f. for A154372. - Peter Bala, Oct 09 2011

From the e.g.f., the row polynomials R(n,t) satisfy the recursion R(n,t) = 1 + t*sum {k = 0..n-1} n!/(k!*(n-k-1)!)*R(n-k-1,t). The polynomials 1/n!*R(n,x) are the polynomials P(n,x) of A152650. Row sums of triangle are A072597. - Peter Bala, Oct 09 2011

EXAMPLE

a(8)=18. Then a(6)+a(7)+a(8)+a(9)=A072597(3)=37.

From Omar E. Pol, Jan 06 2009: (Start)

T(2,3)=960 because (3+1)^2*(2+3)!/2! = 16*120/2 = 960.

Array begins:

1, 1, 2, 6, 24, 120,

1, 4, 18, 96, 600,

1, 12, 108, 960,

1, 32, 540,

1, 80,

1,

(End)

MATHEMATICA

len = 45; m = 1 + Ceiling[Sqrt[len]]; Sort[Flatten[#, 1] &[MapIndexed[ {(2 + #2[[1]]^2 + (#2[[2]] - 1)*#2[[2]] + #2[[1]]*(2*#2[[2]] - 3))/ 2, #1}& , Table[(k + 1)^n*(n + k)!/n!, {n, 0, m}, {k, 0, m}], {2}]]][[All, 2]][[1 ;; len]] (* From Jean-Fran├žois Alcover, May 27 2011 *)

PROG

(Sage)

def A152818_row(n):

    R.<x> = ZZ[]

    P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))

    return P.coefficients()

for n in (0..5): print A152818_row(n)  # Peter Luschny, May 03 2013

(PARI) T(n, k) = (k+1)^n*(n+k)!/n! \\ Charles R Greathouse IV, Sep 10 2016

CROSSREFS

Cf. A000012, A000142, A001563, A001787, A006043, A006044, A055533, A072597 (row sums), A152650, A154372.

Sequence in context: A049429 A183158 A174005 * A302235 A242861 A109244

Adjacent sequences:  A152815 A152816 A152817 * A152819 A152820 A152821

KEYWORD

nonn,tabl

AUTHOR

Paul Curtz, Dec 13 2008

EXTENSIONS

Better definition, extended and edited by Omar E. Pol and N. J. A. Sloane, Jan 05 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 26 06:48 EDT 2019. Contains 321481 sequences. (Running on oeis4.)