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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A241171 Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n. 21
1, 1, 6, 1, 30, 90, 1, 126, 1260, 2520, 1, 510, 13230, 75600, 113400, 1, 2046, 126720, 1580040, 6237000, 7484400, 1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400, 1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000, 1, 131070, 96461910, 8203431600, 196556560200, 1882311631200, 8266953895200, 16672848192000, 12504636144000 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,k) gives the number of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. An example is given below. Cf. A019538 and A156289. - Peter Bala, Aug 20 2014

REFERENCES

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.

S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126.

S. A. Joffe, Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 48 (1917-1920), 193-271.

LINKS

Table of n, a(n) for n=1..45.

FORMULA

T(n,k) = 0 if k<=0 or k>n, = 1 if k=1, otherwise T(n,k) = k*(2*k-1)*T(n-1,k-1)+k^2*T(n-1,k).

Related to Euler numbers A000364 by A000364(n) = (-1)^n*Sum_{k=1..n}(-1)^k*T(n,k). For example, A000364(3) = 61 = 90 - 30 + 1.

From Peter Bala, Aug 20 2014: (Start)

T(n,k) = 1/(2^(k-1))*sum {j = 1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).

T(n,k) = k!*A156289(n,k) = k!*(2*k-1)!!*A036969.

E.g.f.: A(t,z) := 1/( 1 - t*(cosh(z) - 1) ) = 1 + t*z^2/2! + (t + 6*t^2)*z^4/4! + (t + 30*t^2 + 90*t^3)*z^6/6! + ... satisfies the partial differential equation d^2/dz^2(A) = D(A), where D = t^2*(2*t + 1)*d^2/dt^2 + t*(5*t + 1)*d/dt + t.

Hence the row polynomials R(n,t) satisfy the differential equation R(n+1,t) = t^2*(2*t + 1)*R''(n,t) + t*(5*t + 1)*R'(n,t) + t*R(n,t) with R(0,t) = 1, where ' indicates differentiation w.r.t. t. This is equivalent to the above recurrence equation.

Recurrence for row polynomials: R(n,t) = t*( sum {k = 1..n} binomial(2*n,2*k)*R(n-k,t) ) with R(0,t) := 1.

Row sums equal A094088(n) for n >= 1.

A100872(n) = 1/2*R(n,2). (End)

EXAMPLE

Triangle begins:

1,

1, 6,

1, 30, 90,

1, 126, 1260, 2520,

1, 510, 13230, 75600, 113400,

1, 2046, 126720, 1580040, 6237000, 7484400,

1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400,

1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000,

...

From Peter Bala, Aug 20 2014: (Start)

Row 2: [1,6]

k  Ordered set partitions of {1,2,3,4} into k blocks    Number

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1   {1,2,3,4}                                             1

2   {1,2}{3,4}, {3,4}{1,2}, {1,3}{2,4}, {2,4}{1,3},       6

    {1,4}{2,3}, {2,3}{1,4}

(End)

MAPLE

T:=proc(n, k) option remember;

if k = 0 or k > n then 0

elif k=1 then 1

else k*(2*k-1)*T(n-1, k-1)+k^2*T(n-1, k); fi;

end;

for n from 1 to 12 do lprint([seq(T(n, k), k=1..n)]); od:

PROG

(Sage)

@cached_function

def T(n, k):

    if n == 0 and k == 0: return 1

    if k < 0 or k > n: return 0

    return (2*k^2 - k)*T(n - 1, k - 1) + k^2*T(n - 1, k)

for n in (1..6): print [T(n, k) for k in (1..n)] # Peter Luschny, Sep 06 2017

CROSSREFS

Case m=2 of the polynomials defined in A278073.

Cf. A000680 (diagonal), A094088 (row sums), A000364 (alternating row sums), A281478 (central terms).

Diagonals give A002446, A213455, A241172, A002456.

Cf. A019538, A036969, A156289.

Sequence in context: A120101 A178726 A030524 * A051930 A147320 A038255

Adjacent sequences:  A241168 A241169 A241170 * A241172 A241173 A241174

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Apr 22 2014

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 23 05:50 EDT 2018. Contains 316519 sequences. (Running on oeis4.)