1,4

A000670 appears in the first column. A052882 appears in the second column. A000027 and A045943 appear as diagonals. An alternative to calculating the matrix inverse of A154926 is to move the term in the lower right corner to a position in the same column and calculate the determinant instead, which yields the same answer.

Contribution from Peter Bala (pbala(AT)talktalk.net), Jul 01 2009: (Start)

Matrix inverse of (2*I - P), where P is Pascal's triangle and I the

identity matrix. See A162312 for the matrix inverse of (2*P - I) and

some general remarks about arrays of the form M(a) := (I - a*P)^-1

and their connection with weighted sums of powers of integers. The

present array equals 1/2*M(1/2).

(End)

The values in this triangle can be seen as permanents of the Pascal triangle analogous to the method in the Redheffer matrix. The elements satisfy T(n,k)/T(n,k-1)*k=T(n-1,k)/T(n,k)*n which converges to log(2) as n-->infinity and k-->0. More generally to calculate log(x) multiply the negative values in A154926 with 1/(x-1) and calculate the matrix inverse. Then T(n,k)/T(n,k-1)*k and T(n-1,k)/T(n,k)*n in the resulting triangle converge to log(x). [From Mats Granvik (mats.granvik(AT)abo.fi), Aug 11 2009]

This method for calculating log(x) converges faster than the Taylor series when x is greater than 5 or so. See chapter on Taylor series in Spiegel for comparison. [From Mats Granvik (mats.granvik(AT)abo.fi), Aug 11 2009]

R. B. Nelsen, Problem E3062: Amer. Math. Monthly, Vol. 94, No. 4 (Apr., 1987), 376-377. [From Peter Bala (pbala(AT)talktalk.net), Jul 01 2009]

R. B. Nelsen and H. Schmidt, Jr., Chains in Power Sets, Mathematics Magazine, Vol. 64, No. 1 (Feb., 1991), 23-31. [From Peter Bala (pbala(AT)talktalk.net), Jul 01 2009]

Murray R. Spiegel, Mathematical handbook, Schaum's Outlines, p. 111. [From Mats Granvik (mats.granvik(AT)abo.fi), Aug 11 2009]

Contribution from Peter Bala (pbala(AT)talktalk.net), Jul 01 2009: (Start)

TABLE ENTRIES

(1)... T(n,k) = binomial(n,k)*A000670(n-k).

GENERATING FUNCTION

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

PROPERTIES OF THE ROW POLYNOMIALS

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

appear in the study of the poset of power sets [Nelsen and Schmidt].

The first few values are R_0(x) = 1, R_1(x) = 1+x, R_2(x) = 3+2*x+x^2

and R_3(x) = 13+9*x+3*x^2+x^3.

The row polynomials may be recursively computed by means of

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

Explicit formulas include

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

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

*(x+k)^n,

and

(6)... R_n(x) = sum {j = 0..n} sum{k = j..n} k!*Stirling2(n,k)

*binomial(x,k-j).

SUMS OF POWERS OF INTEGERS

The row polynomials satisfy the difference equation

(7)... 2*R_m(x) - R_m(x+1) = x^m,

which easily leads to the evaluation of the weighted sums of powers

of integers

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

For example, m = 2 gives

(9)... sum {k = 1..n-1} (1/2)^k*k^2 = 6 - (1/2)^(n-1)*(n^2+2*n+3).

More generally we have

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

RELATIONS WITH OTHER SEQUENCES

Sequences in the database given by particular values of the row

polynomials are

(11)... A000670(n) = R_n(0)

(12)... A052841(n) = R_n(-1)

(13)... A000629(n) = R_n(1)

(14)... A007047(n) = R_n(2)

(15)... A080253(n) = 2^n*R_n(1/2).

This last result is the particular case (x = 0) of the result that

the polynomials 2^n*R_n(1/2+x/2) are the row generating polynomials

for A162313.

The above formulas should be compared with those for A162312.

(End)

Contribution from Peter Bala (pbala(AT)talktalk.net), Jul 01 2009: (Start)

Triangle begins

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

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

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

0..|.....1

1..|.....1.....1

2..|.....3.....2.....1

3..|....13.....9.....3.....1

4..|....75....52....18.....4.....1

5..|...541...375...130....30.....5.....1

6..|..4683..3246..1125...260....45.....6.....1

...

(End)

Contribution from Mats Granvik (mats.granvik(AT)abo.fi), Aug 11 2009: (Start)

Row 4 equals 75,52,18,4,1 because permanents of:

1,0,0,0,1..1,0,0,0,0..1,0,0,0,0..1,0,0,0,0..1,0,0,0,0

1,1,0,0,0..1,1,0,0,1..1,1,0,0,0..1,1,0,0,0..1,1,0,0,0

1,2,1,0,0..1,2,1,0,0..1,2,1,0,1..1,2,1,0,0..1,2,1,0,0

1,3,3,1,0..1,3,3,1,0..1,3,3,1,0..1,3,3,1,1..1,3,3,1,0

1,4,6,4,0..1,4,6,4,0..1,4,6,4,0..1,4,6,4,0..1,4,6,4,1

are:

75.........52.........18.........4..........1

(End)

A000629 (row sums), A000670, A007047, A052822 (column 1), A052841 (alt. row sums), A080253, A162312, A162313. [From Peter Bala (pbala(AT)talktalk.net), Jul 01 2009]

Sequence in context: A134090 A132845 A129652 this_sequence A127126 A161133 A112911

Adjacent sequences: A154918 A154919 A154920 this_sequence A154922 A154923 A154924

nonn,tabl

Mats Granvik (mats.granvik(AT)abo.fi), Jan 17 2009