A178252 Triangle T(n,m) read by rows: the numerator of the coefficient [x^m] of the umbral inverse Bernoulli polynomials B^{-1}(n,x), 0 <= m <= n.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 5, 10, 5, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 4, 28, 14, 14, 28, 4, 1, 1, 1, 9, 12, 21, 126, 21, 12, 9, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 11, 55, 165, 66, 77, 66, 165, 55, 11, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1

Offset: 0

Author

Paul Curtz, May 24 2010

Keywords

• nonn
• tabl
• frac

Comments

The fractions A053382(n,m)/A053383(n,m) give the triangle of the coefficients of the Bernoulli polynomials:

1;

-1/2, 1;

1/6, -1, 1;

0, 1/2, -3/2, 1;

-1/30, 0, 1, -2, 1;

0, -1/6, 0, 5/3, -5/2, 1;

1/42, 0, -1/2, 0, 5/2, -3, 1;

The matrix inverse of this triangle defines coefficients of the umbral inverse Bernoulli polynomials B^{-1}(n,x) in row n:

1;

1/2, 1;

1/3, 1, 1;

1/4, 1, 3/2, 1;

1/5, 1, 2, 2, 1;

1/6, 1, 5/2, 10/3, 5/2, 1;

1/7, 1, 3, 5, 5, 3, 1;

1/8, 1, 7/2, 7, 35/4, 7, 7/2, 1;

1/9, 1, 4, 28/3, 14, 14, 28/3, 4, 1;

1/10, 1, 9/2, 12, 21, 126/5, 21, 12, 9/2, 1;

1/11, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1;

The current triangle T(n,m) is the numerator of the entry in row n and column m.

In the majority of cases, T(n,m) = A050169(n,m), but since we use the numerators of the reduced fractions, an integer factor may be missing in this equation.

Umbral composition (e.g., B(.,x)^k = B(k,x)) gives B^(-1)(n,B(.,x)) = x^n = B(n,B^(-1)(.,x)). - Tom Copeland, Aug 25 2015

Examples

			The triangle T(n,k) begins:
n\k 0 1  2  3   4   5   6    7   8   9  10 11 12 13
0:  1
1:  1 1
2:  1 1  1
3:  1 1  3  1
4:  1 1  2  2   1
5:  1 1  5 10   5   1
6:  1 1  3  5   5   3   1
7:  1 1  7  7  35   7   7    1
8:  1 1  4 28  14  14  28    4   1
9:  1 1  9 12  21 126  21   12   9   1
10: 1 1  5 15  30  42  42   30  15   5   1
11: 1 1 11 55 165  66  77   66 165  55  11  1
12: 1 1  6 22  55  99 132  132  99  55  22  6  1
13: 1 1 13 26 143 143 429 1716 429 143 143 26 13  1
... reformatted. - _Wolfdieter Lang_, Aug 25 2015
		

Crossrefs

Cf. A178340 (denominators).

Cf. A074909, A258820, A003989, A053382, A053383.

Programs

Maple

nm := 15 : eM := Matrix(nm,nm) :
for n from 0 to nm-1 do for m from 0 to n do eM[n+1,m+1] :=coeff(bernoulli(n,x),x,m) ; end do: for m from n+1 to nm-1 do eM[n+1,m+1] := 0 ; end do: end do:
eM := LinearAlgebra[MatrixInverse](eM) :
for n from 1 to nm do for m from 1 to n do printf("%a,", numer(eM[n,m])) ; end do: end do: # R. J. Mathar, Dec 21 2010

Mathematica

max = 13; coes = Table[ PadRight[ CoefficientList[ BernoulliB[n, x], x], max], {n, 0, max-1}]; inv = Inverse[coes]; Table[ Take[inv[[n]], n], {n, 1, max}] // Flatten // Numerator (* Jean-François Alcover, Aug 09 2012 *)

PARI

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(numerator(binomial(n+1,k)/(n+1)), ", ");); print(););} \\ after Tom Copeland comment; Michel Marcus, Jul 25 2015

Formula

"Palindromic:" T(n,m+1) = T(n,n-m). T(n,0)=1.

From Tom Copeland, Jun 18 2015: (Start)

The umbral inverse Bernoulli polynomials are Binv(n,x) = [(1+x)^(n+1)-x^(n+1)]/(n+1) with the e.g.f. e^(t*x) * (e^t-1)/t. (See A074909 for more details.) Therefore, T(n,k) is the numerator of the reduced fraction C(n+1,k)/(n+1) for 0 <= k < (n+1).

The reversed rows are presented as the diagonals of A258820.

T(n,k) = A258820(2n-k,n-k) = A003989(n+1,n+1-k) * n! / [ k! (n+1-k)! ], where A003989(j,k) = gcd(j,k). (End)

From Wolfdieter Lang, Aug 26 2015: (Start)

The following refers to the rational triangle TBinv with entries T(n,k)/A178340(n, m), n >= m >= 0.

The inverse of the Bernoulli triangle TB(n, m) with entries A196838(n,m)/A196839(n,m), n >= m >= 0, is the Sheffer triangle (z/(exp(z)-1),z). Therefore, the inverse triangle TBinv is the Sheffer triangle ((exp(z)-1)/z, z). This means that the e.g.f. of the sequence of column m of TBinv ((exp(x)-1)/x)*x^m/m! for m >= 0.

The e.g.f. of the row polynomials of TBinv, called Binv(n, x) = Sum_{m=0..n} TBinv(n,m)*x^m, is gBinv(z,x) = ((exp(z)-1)/z)*exp(x*z) (of the so-called Appell type).

The e.g.f. of the row sums is gBinv(x,1).

The e.g.f. of the alternating row sums is gBinv(x,-1) = (1 - exp(-x))/x.

The e.g.f. of the a-sequence of this Sheffer triangle is 1, and the e.g.f. of the z-sequence is (exp(x) - x -1)/((exp(x) -1)*x). This is the sequence 1/2, -1/12, 0, 1/120, 0, -1/252, 0, 1/240, 0, -1/132, .... For a- and z-sequences of Sheffer triangles and the corresponding recurrences see A006232.

The convolution property of the row polynomials Binv(n, x) is Binv(n, x+y) = Sum_{k=0..n} binomial(n, k)*Binv(n-k, x)*y^n (or with x and y exchanged).

The row polynomials satisfy (d/dx)Binv(n, x) = n*Binv(n-1, x), with Binv(0, x) = 1 (from Meixner's identity).

(End)

Extensions

Redefined based on reduced fractions by R. J. Mathar, Dec 21 2010

The term umbral was added by Tom Copeland, Aug 25 2015