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A282628 Triangle T(n, k) read by rows: row n gives for n >= 0 the coefficients of the exponential numerator polynomial used for the exponential generating function of {Sum_{j=1..m} (1 + 2*j)^n}_{m>=0}. 1
1, 1, 1, 1, 3, 2, 1, 9, 16, 8, 1, 27, 98, 120, 48, 1, 81, 544, 1232, 1152, 384, 1, 243, 2882, 10800, 17760, 13440, 3840, 1, 729, 14896, 87128, 224640, 289920, 184320, 46080, 1, 2187, 75938, 669480, 2544528, 4986240, 5295360, 2903040, 645120, 1, 6561, 384064, 4990112, 26917632, 75204864, 118702080, 107089920, 51609600, 10321920 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

-1,5

COMMENTS

The n = -1 row with T(-1, 0) = 1 has been added in order to have a triangle, and is not used.

For n >= 0 the exponential row polynomials are R(n, t) = Sum_{k=0..n+1} T(n, k)*t^k/k!.

The e.g.f. Eodd(n, t) = Sum_{m >= 0} Sodd(n, m)*t^m/m! with Sodd(n, m) = Sum_{j=0..m} (1+2*j)^n is R(n, t)*exp(t), for n >= 0.

This triangle is the e.g.f. companion of A060187 which gives the coefficients of the row polynomials for the o.g.f.s of {Sodd(n, m)}_{m>=0}, which are

  G(n, x) = Sum_{k=0..n} A060187(n+1,k+1) * x^k / (1 - x)^(n+2), for n >= 0.

The inverse Laplace transform L^[-1] is used to obtain the present triangle from A060187. To accomplish this the following reordering identity is used:

  (Sum_{j=0..n} a(n, j)*x^j)/(1-x)^(n+1) = Sum_{k=0..n} (b(n, k)*x^k/(1-x)^(k+1)), with b(n, k) = Sum_{p=0..min(k,n)} binomial(n-p, k-p)*a(n, p), for n >= 0. This can be proved by multiplying with (1-x)^(n+1) and using the binomial theorem to find first a(n, j) = Sum_{i=0..min(n,j)} (-1)^(j-i)*binomial(n-i, j-i)*b(n,i). This is then inverted by using the binomial identity (5.24) of Graham et al., p. 169, to find b in terms of a.

This leads finally to the inverse Laplace transform formula L^[-1]{(Sum_{j=0..n} a(n, j) * x^j) / (1-x)^(n+1)} = exp(t)*Sum_{k=0..n} b(n, k)*t^k/k!, for n >= 0, with the given expression for b(n, k). This is then applied on the o.g.f. G(n, x) given above.

On can obtain Sodd(n, m) in two ways from S(n,m) = Sum_{j=1..n} j^n by bisection of the j sum: Sodd(n, m) = S(n, 2*(m+1)) - 2^n*S(n, m+1) = S(n, 2*m+1) - 2^n*S(n, m).

The first columns of the triangle are A000012, A000244, 2*A005059, 8*A017389, 48*A028060, ...

The diagonal for n >= 0 is A000165. This is compatible with the second formula for T(n, k) given below.

For the generated sequences for n = 0..4 see A000027, A000290, A000447, A002593, A002309.

(with different offsets).

REFERENCES

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd ed.; Addison-Wesley, 1994, p. 169, eq. (5.24).

LINKS

Table of n, a(n) for n=-1..53.

FORMULA

T(n, k) = 0 for k > n+1, T(-1, 0) = 1, and

  T(n ,k) = Sum_{j=0..min(n+1,k)} binomial(n+1-j,k-j)*A060187(n+1,j+1), for n >= 0 and k = 0..n+1.

T(n, k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k-1, j-1) * (2*j+1)^n, for n >= 0 and k = 0..n+1 (if one puts here binomial(-1, -1) = 1).

EXAMPLE

The triangle T(n, k) begins (row n=-1 is not used):

n\k    0    1      2       3        4        5         6          7        8

-1:    1

+0:    1    1

+1:    1    3      2

+2:    1    9     16       8

+3:    1   27     98     120       48

+4:    1   81    544    1232     1152      384

+5:    1  243   2882   10800    17760    13440      3840

+6:    1  729  14896   87128   224640   289920    184320      46080

+7:    1 2187  75938  669480  2544528  4986240   5295360    2903040   645120

...

row n=8: 1 6561 384064 4990112 26917632 75204864 118702080  107089920 51609600 10321920,

row n=9: 1 19683 1933442 36467040 272199360 1042594560 2295175680 3030773760 2376622080 1021870080 185794560. ...

n = 0: Eodd(0, t) = R(0, t)*exp(t) =  (1 + 1*t)*exp(t). G(0, x) = 1/(1 - x)^2.

n = 2: Eodd(3, t) = (1 + 9*t + 16*t^2/2! + 8*t^3/3!)*exp(t), G(2, x) = (1 + 6*x + x^2)/(1 - x)^4.

MATHEMATICA

Table[Sum[(-1)^(k - j) Binomial[k - 1, j - 1] (2 j + 1)^n, {j, 0, k}], {n, -1, 8}, {k, 0, n + 1}] // Flatten (* Michael De Vlieger, Mar 17 2017 *)

PROG

(PARI) {for(n=-1, 8, for(k=0, n + 1, print1(if(k==0, 1, sum(j=0, k, (-1)^(k - j) * binomial(k - 1, j - 1) * (2*j + 1)^n)), ", "); ); print(); ); } \\ Indranil Ghosh, Mar 18 2017

CROSSREFS

Cf. A060187, A000165 (diagonal).

Columns: A000012, A000244, 2*A005059, 8*A017389, 48*A028060.

Cf. Generated sequences (with offset differing): A000027, A000290, A000447, A002593, A002309.

Sequence in context: A121581 A162976 A106338 * A262554 A129964 A267328

Adjacent sequences:  A282625 A282626 A282627 * A282629 A282630 A282631

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Mar 14 2017

STATUS

approved

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Last modified February 24 18:24 EST 2018. Contains 299628 sequences. (Running on oeis4.)