A186024 Inverse of eigentriangle of triangle A085478.
1, -1, 1, -1, -1, 1, -1, -3, -1, 1, -1, -6, -5, -1, 1, -1, -10, -15, -7, -1, 1, -1, -15, -35, -28, -9, -1, 1, -1, -21, -70, -84, -45, -11, -1, 1, -1, -28, -126, -210, -165, -66, -13, -1, 1, -1, -36, -210, -462, -495, -286, -91, -15, -1, 1, -1, -45, -330, -924, -1287, -1001, -455, -120, -17, -1, 1
Offset: 0
Examples
Triangle begins 1, -1, 1, -1, -1, 1, -1, -3, -1, 1, -1, -6, -5, -1, 1, -1, -10, -15, -7, -1, 1, -1, -15, -35, -28, -9, -1, 1, -1, -21, -70, -84, -45, -11, -1, 1, -1, -28, -126, -210, -165, -66, -13, -1, 1, -1, -36, -210, -462, -495, -286, -91, -15, -1, 1, -1, -45, -330, -924, -1287, -1001, -455, -120, -17, -1, 1
Formula
T(n,k)=if(k
A220671 Coefficient array for powers of x^2 of polynomials appearing in a generalized Melham conjecture on alternating sums of fifth powers of Chebyshev S polynomials with odd indices.
-14, 15, -20, 8, -1, 55, -170, 221, -153, 59, -12, 1, 115, -670, 1773, -2696, 2549, -1538, 589, -138, 18, -1, 195, -1850, 8215, -21530, 36330, -41110, 31865, -17080, 6314, -1579, 255, -24, 1, 295, -4150, 27735, -110795, 289540, -518290, 654595, -595805, 396316, -193906, 69641, -18129, 3327, -408, 30, -1
Offset: 1
Comments
The row lengths sequence is 3*n + 1 = A016777(n).
For the generalized Melham conjecture and links to the references concerned with the Melham conjecture on sums of fifth powers of even-indexed Fibonacci numbers see a comment under A220670.
Here the conjecture is considered for m=2 (fifth powers): H(2,n,x^2):= product(tau(j,x), j=0..2) * sum(((-1)^k)*(S(2*k-1,x)/x)^5, k=0..n) / (P(n,x^2)^2), with P(n,x^2):= (1 - (-1)^n*S(2*n,x))/x^2. For tau(j,x):= 2*T(2*j+1,x/2)/x, with Chebyshev's T polynomials see a Oct 23 2012 comment on A111125. For the polynomials P see signed A109954. The conjecture is that H(2,n,x^2) is an integer polynomial of degree 3*n: H(2,n,x^2) = sum(a(n,p)*x^(2*p), p=0..3*n), n >= 1.
If one puts x = i (the imaginary unit) one obtains the original Melham conjecture for Fibonacci numbers F = A000045.
H(2,n,-1) = +44*sum(F(2*k)^5,k=0..n)/(1+F(2*n+1))^2, n>=1, which is conjectured to be -14 - 3*y(n) + 8*y(n)^2 + 4*y(n)^3, with y(n):=F(2*n+1) (see row m=2 of A217475).
It is conjectured that H(2,n,x^2) = h(2,n,x^2) - 3*z(n) + 8*z(n)^2 + 4*z(n)^3, with z(n):= ((-1)^n)*S(2*n,x), with h an integer polynomial of degree 3*n. See A220672 for the coefficients of h(2,n,x^2) for n = 1..5. Because h(2,n,-1) = -14 by the usual Melham conjecture, we put h(2,0,x^2) = -14.
Examples
The array a(n,p) begins: n\p 0 1 2 3 4 5 6 7 8 9 10 11 12 0: -14 1: 15 -20 8 -1 2: 55 -170 221 -153 59 -12 1 3: 115 -670 1773 -2696 2549 -1538 589 -138 18 -1 4: 195 -1850 8215 -21530 36330 -41110 31865 -17080 6314 -1579 255 -24 1 ... Row n=5: [295, -4150, 27735, -110795, 289540, -518290, 654595, -595805, 396316, -193906, 69641, -18129, 3327, -408, 30, -1], Row n=6: [415, -8120, 76118, -429531, 1599441, -4125672, 7621983, -10350335, 10539787, -8164410, 4853792, -2222153, 781514, -209172, 41823, -6047, 597, -36, 1].
Formula
a(n,p) = [x^(2*p)] H(2,n,x^2), n>=1, with H(2,n,x^2) defined in a comment above. a(0,0) has been put to -14 ad hoc.
A220672 Coefficients of powers of x^2 of polynomials, called h(2,n,x^2), appearing in a conjecture on alternating sums of fifth powers of odd-indexed Chebyshev S polynomials stated in A220671.
-14, 6, 5, -12, 3, 46, -95, 16, 75, -69, 24, -3, 106, -520, 928, -607, -351, 894, -651, 234, -42, 3, 186, -1600, 5840, -11355, 11005, -1110, -9615, 11580, -6906, 2433, -513, 60, -3, 286, -3775, 22360, -75595, 153515, -177565, 77115, 84495, -171324, 145302, -75831, 26235, -6057, 900, -78, 3
Offset: 0
Comments
The row lengths sequence for this irregular triangle is 3*n+1 = A016777(n).
A generalized Melham conjecture involving fifth powers (m=2) of odd-indexed Chebyshev S polynomials (see A049310) is H(2,n,x^2):= (x^2-3)*(x^4-5*x^2+5)*sum(((-1)^k)*(S(2*k-1,x)/x)^(2*m+1), k=0..n)/((1 - (-1)^n*S(2*n,x))/x^2)^2 = h(2,n,x^2) - 3*z(n) + 8*z(n)^2 + 4*z(n)^3, with z(n):= ((-1)^n)*S(2*n,x), and h an integer polynomial of degree 3*n.
The present array a(n,p) appears as h(2,n,x^2) = sum(a(n,p)*x^(2*p),p=0..3*n), n >= 1. The entry a(0,0) := -14 has been used because, in accordance with the original Melham conjecture (see a comment on A220671), h(2,n,i^2), with the imaginary unit i, is conjectured to be -14, for all n >= 1.
[-14, -3, 8, 4] is row m=2 of A217475.
Examples
The array a(n,p) begins: n\p 0 1 2 3 4 5 6 7 8 9 0: -14 1: 6 5 -12 3 2: 46 -95 16 75 -69 24 -3 3: 106 -520 928 -607 -351 894 -651 234 -42 3 ... Row n=4: [186, -1600, 5840, -11355, 11005, -1110, -9615, 11580, -6906, 2433, -513, 60, -3]; Row n=5: [286, -3775, 22360, -75595, 153515, -177565, 77115, 84495, -171324, 145302, -75831, 26235, -6057, 900, -78, 3]. Thus the conjecture is true at least for n=1..5.
Formula
a(n,p) = [x^(2p)] h(0,2,n,x^2), with the polynomial h defined above in a comment. The conjecture is that h is an integer polynomial of degree 3n in x^2.
Comments