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A178395
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Triangle T(n,m) read by rows: the numerator of the coefficient [x^m] of the inverse Euler polynomial E^{-1}(n,x), 0 <= m <= n.
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1
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1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 15, 10, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 4, 14, 28, 35, 28, 14, 4, 1, 1, 9, 18, 42, 63, 63, 42, 18, 9, 1, 1, 5, 45, 60, 105, 126, 105, 60, 45, 5, 1, 1, 11, 55, 165, 165, 231, 231, 165, 165, 55, 11, 1, 1, 6, 33, 110, 495, 396, 462, 396, 495, 110, 33, 6, 1
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OFFSET
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0,8
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COMMENTS
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The triangle of fractions A060096(n,m)/A060097(n,m) contains the coefficients of the Euler Polynomial E(n,x) in row n. The matrix inverse of this triangle is
1;
1/2, 1;
1/2, 1, 1;
1/2, 3/2, 3/2, 1;
1/2, 2, 3, 2, 1;
1/2, 5/2, 5, 5, 5/2, 1;
and defines inverse Euler polynomials E^{-1}(n,x) assuming that row n and column m contain the coefficient [x^m] E^{-1}(n,x). The column m=0 is 1 if n=0, otherwise 1/2.
The current triangle T(n,m) shows the numerator of [x^m] E^{-1}(n,x).
Numerators of exponential Riordan array [(1+exp(x))/2,x]. Central coefficients T(2n,n) are A088218. - Paul Barry, Sep 07 2010
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LINKS
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FORMULA
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T(n,0) = 1.
T(n,m) = T(n,n-m).
Number triangle T(n,k) = [k<=n]*numerator((C(n,k) + C(0,n-k))/2). - Paul Barry, Sep 07 2010
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EXAMPLE
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Triangle begins
1;
1, 1;
1, 1, 1;
1, 3, 3, 1;
1, 2, 3, 2, 1;
1, 5, 5, 5, 5, 1;
1, 3, 15, 10, 15, 3, 1;
1, 7, 21, 35, 35, 21, 7, 1;
1, 4, 14, 28, 35, 28, 14, 4, 1;
1, 9, 18, 42, 63, 63, 42, 18, 9, 1;
1, 5, 45, 60, 105, 126, 105, 60, 45, 5, 1; (End)
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MAPLE
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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(euler(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("%d, ", numer(eM[n, m])) ; end do: end do: # R. J. Mathar, Dec 21 2010
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MATHEMATICA
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(* The function RiordanArray is defined in A256893. *)
rows = 13;
R = RiordanArray[(1 + E^#)/2&, #&, rows, True];
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PROG
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(PARI) T(n, k)=numerator((binomial(n, k)+binomial(0, n-k))/2);
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print());
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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