Real-rooted atlas

A121574 → basis A013611 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-100\,n + 800\,n x - 1536\,n x^{2} - 512\,n x^{3} + 1024\,n x^{4} + 688\,n^{2} - 64\,n^{2} x + 1280\,n^{2} x^{2} + 1024\,n^{2} x^{3} - 192\,n^{3} + 768\,n^{3} x\right)\,P_{n}(x) = \left(-732 - 6128\,x - 13440\,x^{2} - 512\,x^{3} + 9216\,x^{4} + 4096\,x^{5} + 1648\,n + 10304\,n x + 16640\,n x^{2} + 7168\,n x^{3} + 3392\,n^{2} + 11008\,n^{2} x + 1024\,n^{2} x^{2} - 4096\,n^{2} x^{3} - 1536\,n^{3} - 3072\,n^{3} x\right)\,P_{n-1}(x) + \left(-91 - 696\,x - 144\,x^{2} + 9984\,x^{3} + 25344\,x^{4} + 18432\,x^{5} + 4096\,x^{6} + 260\,n + 3088\,n x + 13184\,n x^{2} + 24064\,n x^{3} + 17408\,n x^{4} + 4096\,n x^{5} + 624\,n^{2} + 4416\,n^{2} x + 8448\,n^{2} x^{2} + 3072\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
2386016114
3173492368641097
472930524096198425610117
529281629632400279689984102490600
61138979320211008269760170496481284096794197
743291362596121358420801921942784965632225280163846849743

All 15 rows

Technical details
Node id
n-94e80b396f849c9f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3931adb004cd6925

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      7,       4;
  2:     38,      60,       16;
  3:    173,     492,      368,       64;
  4:    729,    3052,     4096,     1984,       256;
  5:   2928,   16296,    32400,    27968,      9984,      1024;
  6:  11389,   79320,   211008,   269760,    170496,     48128,     4096;
  7:  43291,  362596,  1213584,  2080192,   1942784,    965632,   225280,    16384;
  8: 161726, 1584084,  6404128, 13846592,  17339136,  12700672,  5193728,  1032192,   65536;
  9: 596057, 6687396, 31729808, 83075392, 131347968, 128884736, 77467648, 26886144, 4653056, 262144;
%F Formula
The row polynomials satisfy -(192*n^3-688*n^2+100*n-(768*n^3-64*n^2+800*n)*x-(1280*n^2-1536*n)*x^2-(1024*n^2-512*n)*x^3-1024*n*x^4)*P(n,x) = -(1536*n^3-3392*n^2-1648*n+732+(3072*n^3-11008*n^2-10304*n+6128)*x-(1024*n^2+16640*n-13440)*x^2+(4096*n^2-7168*n+512)*x^3-9216*x^4-4096*x^5)*P(n-1,x) + (624*n^2+260*n-91+(4416*n^2+3088*n-696)*x+(8448*n^2+13184*n-144)*x^2+(3072*n^2+24064*n+9984)*x^3+(17408*n+25344)*x^4+(4096*n+18432)*x^5+4096*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(1536*n^3 - 3392*n^2 - 1648*n + 732 + (3072*n^3 - 11008*n^2 - 10304*n + 6128)*x - (1024*n^2 + 16640*n - 13440)*x^2 + (4096*n^2 - 7168*n + 512)*x^3 - 9216*x^4 - 4096*x^5)*P[n - 1, x] + (624*n^2 + 260*n - 91 + (4416*n^2 + 3088*n - 696)*x + (8448*n^2 + 13184*n - 144)*x^2 + (3072*n^2 + 24064*n + 9984)*x^3 + (17408*n + 25344)*x^4 + (4096*n + 18432)*x^5 + 4096*x^6)*D[P[n - 1, x], x])/(-192*n^3 + 688*n^2 - 100*n + (768*n^3 - 64*n^2 + 800*n)*x + (1280*n^2 - 1536*n)*x^2 + (1024*n^2 - 512*n)*x^3 + 1024*n*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(1536*n^3 - 3392*n^2 - 1648*n + 732 + (3072*n^3 - 11008*n^2 - 10304*n + 6128)*x - (1024*n^2 + 16640*n - 13440)*x^2 + (4096*n^2 - 7168*n + 512)*x^3 - 9216*x^4 - 4096*x^5)*v[i - 1] + (624*n^2 + 260*n - 91 + (4416*n^2 + 3088*n - 696)*x + (8448*n^2 + 13184*n - 144)*x^2 + (3072*n^2 + 24064*n + 9984)*x^3 + (17408*n + 25344)*x^4 + (4096*n + 18432)*x^5 + 4096*x^6)*deriv(v[i - 1], x))/(-192*n^3 + 688*n^2 - 100*n + (768*n^3 - 64*n^2 + 800*n)*x + (1280*n^2 - 1536*n)*x^2 + (1024*n^2 - 512*n)*x^3 + 1024*n*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.