Real-rooted atlas

A099040 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 6\,x - 32\,x^{2} + 32\,x^{4} + 54\,n x + 30\,n x^{2} - 100\,n x^{3} - 88\,n x^{4} + 78\,n^{2} x^{2} + 64\,n^{2} x^{3} - 68\,n^{2} x^{4} - 48\,n^{2} x^{5} + 36\,n^{3} x^{3} + 24\,n^{3} x^{4} - 16\,n^{3} x^{5} + 4\,n^{4} x^{4}\right)\,P_{n}(x) = \left(18 + 132\,x + 344\,x^{2} + 304\,x^{3} - 192\,x^{4} - 512\,x^{5} - 256\,x^{6} + 108\,n x + 588\,n x^{2} + 1016\,n x^{3} + 272\,n x^{4} - 832\,n x^{5} - 576\,n x^{6} + 156\,n^{2} x^{2} + 656\,n^{2} x^{3} + 744\,n^{2} x^{4} - 80\,n^{2} x^{5} - 384\,n^{2} x^{6} + 72\,n^{3} x^{3} + 224\,n^{3} x^{4} + 128\,n^{3} x^{5} - 64\,n^{3} x^{6} + 8\,n^{4} x^{4} + 16\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(18 + 132\,x + 260\,x^{2} - 40\,x^{3} - 576\,x^{4} - 544\,x^{5} - 192\,x^{6} + 108\,n x + 504\,n x^{2} + 600\,n x^{3} - 336\,n x^{4} - 992\,n x^{5} - 448\,n x^{6} + 156\,n^{2} x^{2} + 584\,n^{2} x^{3} + 504\,n^{2} x^{4} - 240\,n^{2} x^{5} - 320\,n^{2} x^{6} + 72\,n^{3} x^{3} + 208\,n^{3} x^{4} + 96\,n^{3} x^{5} - 64\,n^{3} x^{6} + 8\,n^{4} x^{4} + 16\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121618
2680144230
316320115210242512
4441152619211264640025052
5120390427936768008960036864235224
6328127361137604188167424006266882007042115432
7896404484331521998848476160061194244014080104857618417024

All 15 rows

Technical details
Node id
n-e16a5c5bc932c32c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5b508ded3910d61e

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:    2,     16;
  2:    6,     80,     144;
  3:   16,    320,    1152,     1024;
  4:   44,   1152,    6192,    11264,      6400;
  5:  120,   3904,   27936,    76800,     89600,     36864;
  6:  328,  12736,  113760,   418816,    742400,    626688,    200704;
  7:  896,  40448,  433152,  1998848,   4761600,   6119424,   4014080,   1048576;
  8: 2448, 125952, 1572480,  8724480,  26118400,  45195264,  45158400,  24117248,   5308416;
  9: 6688, 386304, 5508864, 35708928, 128780800, 280498176, 377724928, 307232768, 138018816, 26214400;
%F Formula
The row polynomials satisfy (9+(54*n-6)*x+(78*n^2+30*n-32)*x^2+(36*n^3+64*n^2-100*n)*x^3+(4*n^4+24*n^3-68*n^2-88*n+32)*x^4-(16*n^3+48*n^2)*x^5)*P(n,x) = (18+(108*n+132)*x+(156*n^2+588*n+344)*x^2+(72*n^3+656*n^2+1016*n+304)*x^3+(8*n^4+224*n^3+744*n^2+272*n-192)*x^4+(16*n^4+128*n^3-80*n^2-832*n-512)*x^5-(64*n^3+384*n^2+576*n+256)*x^6)*P(n-2,x) + (18+(108*n+132)*x+(156*n^2+504*n+260)*x^2+(72*n^3+584*n^2+600*n-40)*x^3+(8*n^4+208*n^3+504*n^2-336*n-576)*x^4+(16*n^4+96*n^3-240*n^2-992*n-544)*x^5-(64*n^3+320*n^2+448*n+192)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((18 + (108*n + 132)*x + (156*n^2 + 588*n + 344)*x^2 + (72*n^3 + 656*n^2 + 1016*n + 304)*x^3 + (8*n^4 + 224*n^3 + 744*n^2 + 272*n - 192)*x^4 + (16*n^4 + 128*n^3 - 80*n^2 - 832*n - 512)*x^5 - (64*n^3 + 384*n^2 + 576*n + 256)*x^6)*P[n - 2, x] + (18 + (108*n + 132)*x + (156*n^2 + 504*n + 260)*x^2 + (72*n^3 + 584*n^2 + 600*n - 40)*x^3 + (8*n^4 + 208*n^3 + 504*n^2 - 336*n - 576)*x^4 + (16*n^4 + 96*n^3 - 240*n^2 - 992*n - 544)*x^5 - (64*n^3 + 320*n^2 + 448*n + 192)*x^6)*P[n - 1, x])/(9 + (54*n - 6)*x + (78*n^2 + 30*n - 32)*x^2 + (36*n^3 + 64*n^2 - 100*n)*x^3 + (4*n^4 + 24*n^3 - 68*n^2 - 88*n + 32)*x^4 - (16*n^3 + 48*n^2)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = ((18 + (108*n + 132)*x + (156*n^2 + 588*n + 344)*x^2 + (72*n^3 + 656*n^2 + 1016*n + 304)*x^3 + (8*n^4 + 224*n^3 + 744*n^2 + 272*n - 192)*x^4 + (16*n^4 + 128*n^3 - 80*n^2 - 832*n - 512)*x^5 - (64*n^3 + 384*n^2 + 576*n + 256)*x^6)*v[i - 2] + (18 + (108*n + 132)*x + (156*n^2 + 504*n + 260)*x^2 + (72*n^3 + 584*n^2 + 600*n - 40)*x^3 + (8*n^4 + 208*n^3 + 504*n^2 - 336*n - 576)*x^4 + (16*n^4 + 96*n^3 - 240*n^2 - 992*n - 544)*x^5 - (64*n^3 + 320*n^2 + 448*n + 192)*x^6)*v[i - 1])/(9 + (54*n - 6)*x + (78*n^2 + 30*n - 32)*x^2 + (36*n^3 + 64*n^2 - 100*n)*x^3 + (4*n^4 + 24*n^3 - 68*n^2 - 88*n + 32)*x^4 - (16*n^3 + 48*n^2)*x^5)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.