Real-rooted atlas

A207606 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-216 + 216\,x - 96\,x^{2} - 96\,x^{3} - 140\,n + 432\,n x - 80\,n x^{2} - 64\,n x^{3} + 288\,n^{2} x + 16\,n^{3} + 64\,n^{3} x\right)\,P_{n}(x) = \left(-780 + 392\,x + 272\,x^{2} - 416\,x^{3} - 128\,x^{4} - 358\,n + 1100\,n x + 712\,n x^{2} - 240\,n x^{3} - 64\,n x^{4} + 32\,n^{2} + 672\,n^{2} x + 448\,n^{2} x^{2} + 24\,n^{3} + 112\,n^{3} x + 64\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-135 - 264\,x - 224\,x^{2} - 656\,x^{3} - 400\,x^{4} - 64\,x^{5} + 20\,n + 368\,n x + 784\,n x^{2} + 256\,n x^{3} + 20\,n^{2} + 128\,n^{2} x + 208\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 35 + 38\,x + 8\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
03538881
112222411616478
23991054924312322721
312534356550831367846415101
438261656027648226729520188812882242
5114455940812390013416080960268804416256441425
63370320425051242869368855224026294472128101125122342005
798026679792199661232551363239136203212879788818636822784102412308894
8282267220533874297041419844817019632133385286892032230041646771250688204864186813
98060897007976266549045850217682240240776258565006758421966336637286411468801116164096332506617
10228605021892896928278722302955203720371204118492803197994241750049286666624017098240275968024371281921712769154
116444897674220763154177688733242881595788880203054572818488874241217467392578034432194472960446842886537216528384163848779572117
1218077099205124462104976099232102942566551194896943183785698713908487620077568435112550418232023045490524161142415361528217611386883276844811833369
13504789386175671683432507764114941232482592427075241698658304493843566724381231641629317170432147663380485534622720150814310428668723235323904244121665536227865071454
141404088711842443638110535094444023522525699432391920176837881248234024348096234931779456180634823424106579545600479938201601626697113640467947527080263688085504052101121310721154814165593
Technical details
Node id
n-dbfcdf9df0cf209a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d12b8b9500fb4de1

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:     35,      38,        8;
  1:    122,     224,      116,       16;
  2:    399,    1054,      924,      312,       32;
  3:   1253,    4356,     5508,     3136,      784,       64;
  4:   3826,   16560,    27648,    22672,     9520,     1888,      128;
  5:  11445,   59408,   123900,   134160,    80960,    26880,     4416,      256;
  6:  33703,  204250,   512428,   693688,   552240,   262944,    72128,    10112,     512;
  7:  98026,  679792,  1996612,  3255136,  3239136,  2032128,   797888,   186368,   22784,    1024;
  8: 282267, 2205338,  7429704, 14198448, 17019632, 13338528,  6892032,  2300416,  467712,   50688,   2048;
  9: 806089, 7007976, 26654904, 58502176, 82240240, 77625856, 50067584, 21966336, 6372864, 1146880, 111616, 4096;
%F Formula
The row polynomials satisfy (16*n^3-140*n-216+(64*n^3+288*n^2+432*n+216)*x-(80*n+96)*x^2-(64*n+96)*x^3)*P(n,x) = (24*n^3+32*n^2-358*n-780+(112*n^3+672*n^2+1100*n+392)*x+(64*n^3+448*n^2+712*n+272)*x^2-(240*n+416)*x^3-(64*n+128)*x^4)*P(n-1,x) + (20*n^2+20*n-135+(128*n^2+368*n-264)*x+(208*n^2+784*n-224)*x^2+(64*n^2+256*n-656)*x^3-400*x^4-64*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 35 + 38*x + 8*x^2.
%t Mathematica
P[0, x_] := 35 + 38*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n^3 + 32*n^2 - 358*n - 780 + (112*n^3 + 672*n^2 + 1100*n + 392)*x + (64*n^3 + 448*n^2 + 712*n + 272)*x^2 - (240*n + 416)*x^3 - (64*n + 128)*x^4)*P[n - 1, x] + (20*n^2 + 20*n - 135 + (128*n^2 + 368*n - 264)*x + (208*n^2 + 784*n - 224)*x^2 + (64*n^2 + 256*n - 656)*x^3 - 400*x^4 - 64*x^5)*D[P[n - 1, x], x])/(16*n^3 - 140*n - 216 + (64*n^3 + 288*n^2 + 432*n + 216)*x - (80*n + 96)*x^2 - (64*n + 96)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 35 + 38*x + 8*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n^3 + 32*n^2 - 358*n - 780 + (112*n^3 + 672*n^2 + 1100*n + 392)*x + (64*n^3 + 448*n^2 + 712*n + 272)*x^2 - (240*n + 416)*x^3 - (64*n + 128)*x^4)*v[i - 1] + (20*n^2 + 20*n - 135 + (128*n^2 + 368*n - 264)*x + (208*n^2 + 784*n - 224)*x^2 + (64*n^2 + 256*n - 656)*x^3 - 400*x^4 - 64*x^5)*deriv(v[i - 1], x))/(16*n^3 - 140*n - 216 + (64*n^3 + 288*n^2 + 432*n + 216)*x - (80*n + 96)*x^2 - (64*n + 96)*x^3)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023