A185414
Square array, read by antidiagonals, used to recursively calculate the zigzag numbers A000111.
Original entry on oeis.org
1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 16, 16, 10, 4, 1, 61, 61, 39, 17, 5, 1, 272, 272, 176, 80, 26, 6, 1, 1385, 1385, 903, 421, 145, 37, 7, 1, 7936, 7936, 5200, 2464, 880, 240, 50, 8, 1, 50521, 50521, 33219, 15917, 5825, 1661, 371, 65, 9, 1
Offset: 1
The array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
2, 5, 10, 17, 26, 37, 50, 65, 82, ...;
5, 16, 39, 80, 145, 240, 371, 544, 765, ...;
16, 61, 176, 421, 880, 1661, 2896, 4741, 7376, ...;
61, 272, 903, 2464, 5825, 12336, 23947, 43328, 73989, ...;
272, 1385, 5200, 15917, 41936, 98377, 210320, 416765, ...;
1385, 7936, 33219, 112640, 326965, 840960, 1962191, ...; ...
Examples of the recurrence:
T(4,4) = 80 = (3*T(3,3) + 5*T(3,5))/2 = (3*10 + 5*26)/2;
T(5,3) = 176 = (2*T(4,2) + 4*T(4,4))/2 = (2*16 + 4*80)/2;
T(6,2) = 272 = (1*T(5,1) + 3*T(5,3))/2 = (1*16 + 3*176)/2.
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#A185414 Z := proc(n,x)
description 'zigzag polynomials A147309'
if n = 0 return 1 else return 1/2*x*(Z(n-1,x-1)+Z(n-1,x+1))
end proc:
# values of Z(n,x)/x
for n from 1 to 10 do seq(Z(n,k)/k, k = 1..10);
end do;
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{T(n,k)=if(n==1,1,((k-1)*T(n-1,k-1)+(k+1)*T(n-1,k+1))/2)}
for(n=1,10, for(k=1,10, print1(T(n,k),", ")); print(""))
A290580
E.g.f. W = W(x,m) satisfies: W = E(x*W,m) where E(x,m) = cn(i*x,m) - i*sn(i*x,m), with sn(x,m) and cn(x,m) being Jacobi elliptic functions, read as an irregular triangle of coefficients T(n,k) of x^n*m^k for n>=0 and k=0..[n/2].
Original entry on oeis.org
1, 1, 3, 0, 16, 1, 125, 20, 0, 1296, 364, 1, 16807, 7028, 112, 0, 262144, 148752, 5868, 1, 4782969, 3471192, 250128, 576, 0, 100000000, 89097664, 10020912, 82408, 1, 2357947691, 2503362488, 399379728, 7354688, 2816, 0, 61917364224, 76575071488, 16255733440, 533661360, 1066552, 1, 1792160394037, 2536513162508, 684615750832, 35063521792, 194025728, 13312, 0, 56693912375296, 90532686154752, 30031767680256, 2200207121408, 24852054816, 13053492, 1, 1946195068359375, 3465845396598540, 1376568893633760, 135791393602560, 2630843800320, 4759188480, 61440, 0
Offset: 0
E.g.f. W(x,m) = 1 + (1)*x + (3)*x^2/2! + (16 + m)*x^3/3! +
(125 + 20*m)*x^4/4! + (1296 + 364*m + m^2)*x^5/5! +
(16807 + 7028*m + 112*m^2)*x^6/6! +
(262144 + 148752*m + 5868*m^2 + m^3)*x^7/7! +
(4782969 + 3471192*m + 250128*m^2 + 576*m^3)*x^8/8! +
(100000000 + 89097664*m + 10020912*m^2 + 82408*m^3 + m^4)*x^9/9! +
(2357947691 + 2503362488*m + 399379728*m^2 + 7354688*m^3 + 2816*m^4)*x^10/10! +...
such that W = W(x,m) satisfies:
W = E(x*W,m)
where E(x,m) is an elliptic analog to the exponential function, defined by
E(x,m) = cn(i*x,m) - i*sn(i*x,m).
By Jacobi's imaginary transformation, we have
E(x,m) = (1 + sn(x,1-m)) / cn(x,1-m),
where
E(x,m) = 1 + x + x^2/2! + (m + 1)*x^3/3! + (4*m + 1)*x^4/4! + (m^2 + 14*m + 1)*x^5/5! + (16*m^2 + 44*m + 1)*x^6/6! + (m^3 + 135*m^2 + 135*m + 1)*x^7/7! + (64*m^3 + 912*m^2 + 408*m + 1)*x^8/8! + (m^4 + 1228*m^3 + 5478*m^2 + 1228*m + 1)*x^9/9! + (256*m^4 + 15808*m^3 + 30768*m^2 + 3688*m + 1)*x^10/10! +...
Explicitly,
W(x,m) = (1/x) Series_Reversion( x/E(x,m) ).
As a series of row polynomial coefficients of powers of x,
W(x,m) = Sum_{n>=0} x^n/n! * { [x^n/n!] E(x,m)^(n+1) / (n+1) }.
IRREGULAR TRIANGLE.
This triangle of coefficients in e.g.f. W(x,m) begins:
1 ;
1 ;
3, 0 ;
16, 1 ;
125, 20, 0 ;
1296, 364, 1 ;
16807, 7028, 112, 0 ;
262144, 148752, 5868, 1 ;
4782969, 3471192, 250128, 576, 0 ;
100000000, 89097664, 10020912, 82408, 1 ;
2357947691, 2503362488, 399379728, 7354688, 2816, 0 ;
61917364224, 76575071488, 16255733440, 533661360, 1066552, 1 ;
1792160394037, 2536513162508, 684615750832, 35063521792, 194025728, 13312, 0 ; ...
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/* By definition: */
{ T(n,k) = my(W=1,E=1, S=x,C=1,D=1); for(i=0,n,
S = intformal(C*D +x*O(x^n)) ;
C = 1 - intformal(S*D) ; D = 1 - m*intformal(S*C) ;
E = subst(C - I*S,x,I*x) ) ;
for(i=0,n, W = subst(E,x,x*W));
n!*polcoeff(polcoeff(W, n,x), k,m) }
for(n=0,10, for(k=0,n\2, print1( T(n,k), ", ")); print(""))
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/* Using Jacobi's imaginary transformation: */
{ T(n,k) = my(W=1,E=1, S=x,C=1,D=1); for(i=0,n,
S = intformal(C*D +x*O(x^n)) ;
C = 1 - intformal(S*D) ; D = 1 - m*intformal(S*C) ;
E = subst( (1 + S)/C,m,1-m) ) ;
for(i=0,n, W = subst(E,x,x*W));
n!*polcoeff(polcoeff(W, n,x), k,m) }
for(n=0,10, for(k=0,n\2, print1( T(n,k), ", ")); print(""))
Showing 1-2 of 2 results.
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