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A111593
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Triangle of tanh numbers.
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6
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1, 0, 1, 0, 0, 1, 0, -2, 0, 1, 0, 0, -8, 0, 1, 0, 16, 0, -20, 0, 1, 0, 0, 136, 0, -40, 0, 1, 0, -272, 0, 616, 0, -70, 0, 1, 0, 0, -3968, 0, 2016, 0, -112, 0, 1, 0, 7936, 0, -28160, 0, 5376, 0, -168, 0, 1, 0, 0, 176896, 0, -135680, 0, 12432, 0, -240, 0, 1, 0, -353792, 0, 1805056, 0, -508640, 0, 25872
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Sheffer triangle associated to Sheffer triangle A060081.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
In the umbral calculus (see the S. Roman reference) this triangle would be called associated for (1,Artanh(y)).
Without the n=0 row and m=0 column and unsigned, this is the Jabotinsky triangle A059419.
The inverse matrix of A with elements a(n,m), n,m>=0, is A111594.
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060081, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*tanh(y)).
Exponential Riordan array [1, tanh(x)], inverse of [1, atanh(x)] which is A111594. [From Paul Barry (pbarry(AT)wit.ie), May 30 2010]
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LINKS
| W. Lang, First 10 rows.
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FORMULA
| E.g.f. for column m>=0: ((tanh(x))^m)/m!.
a(n, m)= coefficient of x^n of ((tanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) - (m+1)*m*a(n-1, m+1), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n<m.
T(n,m)=sum(k=0..n-m, binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(n-k-m)*stirling2(n,k+m))/m!, T(0,0)=1. [From Vladimir Kruchinin, Jun 09 2011]
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EXAMPLE
| Binomial convolution of row polynomials: p(3,x)= -2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1, together with those from A060081:
s(3,x)= -5*x+x^3; s(2,x)= -1+x^2, s(1,x)= x, s(0,x)= 1;
therefore -5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = -2*y+y^3 + 3*x*y^2 + 3*(-1+x^2)*y + (-5*x+x^3).
Contribution from Paul Barry (pbarry(AT)wit.ie), May 30 2010: (Start)
Triangle begins
1,
0, 1,
0, 0, 1,
0, -2, 0, 1,
0, 0, -8, 0, 1,
0, 16, 0, -20, 0, 1,
0, 0, 136, 0, -40, 0, 1,
0, -272, 0, 616, 0, -70, 0, 1,
0, 0, -3968, 0, 2016, 0, -112, 0, 1
Production matrix begins
0, 1,
0, 0, 1,
0, -2, 0, 1,
0, 0, -6, 0, 1,
0, 0, 0, -12, 0, 1,
0, 0, 0, 0, -20, 0, 1,
0, 0, 0, 0, 0, -30, 0, 1,
0, 0, 0, 0, 0, 0, -42, 0, 1,
0, 0, 0, 0, 0, 0, 0, -56, 0, 1 (End)
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PROG
| (Maxima)
T(n, m):=if n=0 and m=0 then 1 else sum(binomial(k+m-1, m-1)*(k+m)!*(-1)^(k)*2^(n-k-m)*stirling2(n, k+m), k, 0, n-m)/m!; [From Vladimir Kruchinin, Jun 09 2011]
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CROSSREFS
| Row sums: A003723. Unsigned row sums: A006229.
Sequence in context: A036852 A050327 A075120 * A111594 A203951 A105348
Adjacent sequences: A111590 A111591 A111592 * A111594 A111595 A111596
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KEYWORD
| sign,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 23 2005
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