0,2

Unsigned inverse of A162315.

The row generating polynomials of this triangle may be used to evaluate

the weighted sums of powers of odd numbers

(1)... 1^m + 2*3^m + 4*5^m + ... + 2^n*(2*n+1)^m

and also the sums

(2)... 1^m + (1/2)*3^m + (1/4)*5^m + ... + (1/2)^n*(2*n+1)^m.

See the Formula section below.

We make a few remarks about the general array M(a) := a*P*(I-a*P^2)^-1,

where a <> 1, and its connection with weighted sums of powers of odd

numbers. The present case corresponds to a = 1/2. Compare with the

remarks in A162312.

The array M(a) begins

/

| a/(1-a)

| (a^2+a)/(1-a)^2 ................. a/(1-a)

| (a^3+6*a^2+a)/(1-a)^3 ........... 2*(a^2+a)/(1-a)^2 ... a/(1-a)

(a^4+23*a^3+23*a^2+a)/(1-a)^4 ...

| .

\ .

In the first column we recognise the numerator polynomials as the

Eulerian polynomials of type B. See A060187.

The e.g.f. for this array is

(3)... a*exp((x+1)*t)/(1-a*exp(2*t)) = a/(1-a) +[(a^2+a)/(1-a)^2 + a/(1-a)*x]*t + [(a^3+6*a^2+a)/(1-a)^3 + 2*(a^2+a)*x/(1-a)^2 + a/(1-a)*x^2]*t^2/2! + ....

The row polynomials P_m(x), which depend on a, satisfy the difference

equation

(4)... P_m(x) - a*P_m(x+2) = a*(x+1)^m.

for m >= 1.

The first few values are

P_0(x) = a/(1-a), P_1(x) = a*x/(1-a) + (a^2+a)/(1-a)^2 and

P_2(x) = a*x^2/(1-a) + 2*(a^2+a)*x/(1-a)^2 + (a^3+6*a^2+a)/(1-a)^3.

Using (4) leads to the evaluations of the weighted sums of powers of

even and odd positive integers:

(5)... sum {k = 1..n} a^k*(2*k)^m = P_m(1) - a^n*P_m(2*n+1)

and

(6)... sum {k = 1..n} a^k*(2*k-1)^m = P_m(0) - a^n*P_m(2*n),

with m = 0,1,2,... and a <> 1.

In the remaining case a = 1 the sums are evaluated in terms of the

Bernoulli polynomials.

TABLE ENTRIES

(1)... T(n,k) = binomial(n,k)*A080253(n-k).

GENERATING FUNCTION

(2)... exp((x+1)*t)/(2-exp(2*t)) = 1 + (3+x)*t + (17+6*x+x^2)*t^2/2!

+ ....

The e.g.f. can also be written as

(3)... exp(x*t)*G(t), where G(t) = exp(t)/(2-exp(2*t)) is the e.g.f.

for A080253.

ROW GENERATING POLYNOMIALS

The row generating polynomials R_n(x) form an Appell sequence. The

first few values are

R_0(x) = 1, R_1(x) = 3 + x, R_2(x) = 17 + 6*x + x^2 and

R_3(x) = 147 + 51*x + 9*x^2 + x^3.

The row polynomials may be recursively computed by means of

(4)... R_n(x) = (x+1)^n + sum {k=0..n-1} 2^(n-k)*binomial(n,k)*R_k(x).

An explicit formula is

(5)... R_n(x) = sum {j = 0..n} sum {k = 0..j} (-1)^(j-k)*binomial(j,k)

*(x+2*k+1)^n).

There is also a representation as an infinite series

(6)... R_n(x) = 1/2*sum {k = 0..inf}(1/2)^k*(x+2*k+1)^n.

SUMS OF POWERS OF INTEGERS

The row polynomials satisfy the difference equation

(7)... 2*R_n(x) - R_n(x+2) = (x+1)^n,

and hence may be used to evaluate the weighted sums of powers of

odd integers

(8)... sum {k=0..n-1} (1/2)^k*(2*k+1)^m = 2*R_m(0)-1/2^(n-1)*R_m(2*n)

as well as

(9)... sum {k=0..n-1} 2^k*(2*k+1)^m = (-1)^m*(2^n*R_m(-2*n)-R_m(0)).

For example, m = 2 gives

(10)... sum {k=0..n-1} (1/2)^k*(2*k+1)^2 = 34-2^(1-n)*(4*n^2+12*n+17)

and

(11)... sum {k = 0..n-1} 2^k*(2*k+1)^2 = 2^n*(4*n^2 - 12*n + 17)-17.

RELATIONS WITH OTHER SEQUENCES

(12)... Row sums = [1,4,24,208,2400,...] = 2^n*A000629(n) = A162314(n).

(13)... Column 0 = [1,3,17,147,1697,...] = A080253.

The identity

(14)... R_n(2*x-1) = 2^n*P_n(x),

where P_n(x) are the row generating polynomials of A154921, provides

a surprising connection between (6) and the result

(15)... sum {k = 0..n-1} (1/2)^k*k^m = 2*P_m(0) - (1/2)^(n-1)*P_m(n).

Triangle begins

n\k|.......0.......1......2......3......4......5......6

=======================================================

0..|.......1

1..|.......3.......1

2..|......17.......6......1

3..|.....147......51......9......1

4..|....1697.....588....102.....12......1

5..|...24483....8485...1470....170.....15......1

6..|..423857..146898..25455...2940....255.....18......1

...

A000629, A007318, A060187, A080253 (column 0), A154921, A162312, A162314 (row sums), A162315 (unsigned inverse).

Sequence in context: A048159 A123527 A096611 this_sequence A060281 A151918 A089974

Adjacent sequences: A162310 A162311 A162312 this_sequence A162314 A162315 A162316

easy,nonn,tabl

Peter Bala (pbala(AT)talktalk.net), Jul 01 2009