Real-rooted atlas

A063967 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 60\,x + 16\,x^{2} + 128\,x^{3} + 32\,x^{4} - 32\,x^{5} + 54\,n x - 126\,n x^{2} - 120\,n x^{3} + 104\,n x^{4} + 48\,n x^{5} + 78\,n^{2} x^{2} - 44\,n^{2} x^{3} - 116\,n^{2} x^{4} + 36\,n^{3} x^{3} + 8\,n^{3} x^{4} - 16\,n^{3} x^{5} + 4\,n^{4} x^{4}\right)\,P_{n}(x) = \left(18 + 24\,x - 88\,x^{2} - 128\,x^{3} + 64\,x^{4} + 128\,x^{5} + 108\,n x + 276\,n x^{2} - 80\,n x^{3} - 576\,n x^{4} - 352\,n x^{5} + 156\,n^{2} x^{2} + 440\,n^{2} x^{3} + 120\,n^{2} x^{4} - 368\,n^{2} x^{5} - 192\,n^{2} x^{6} + 72\,n^{3} x^{3} + 192\,n^{3} x^{4} + 64\,n^{3} x^{5} - 64\,n^{3} x^{6} + 8\,n^{4} x^{4} + 16\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(18 + 24\,x - 88\,x^{2} - 128\,x^{3} + 64\,x^{4} + 128\,x^{5} + 108\,n x + 192\,n x^{2} - 352\,n x^{3} - 752\,n x^{4} - 288\,n x^{5} + 156\,n^{2} x^{2} + 368\,n^{2} x^{3} - 72\,n^{2} x^{4} - 432\,n^{2} x^{5} - 128\,n^{2} x^{6} + 72\,n^{3} x^{3} + 176\,n^{3} x^{4} + 32\,n^{3} x^{5} - 64\,n^{3} x^{6} + 8\,n^{4} x^{4} + 16\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
221618
3680144230
416320115210242512
5441152619211264640025052
6120390427936768008960036864235224
7328127361137604188167424006266882007042115432
8896404484331521998848476160061194244014080104857618417024

All 15 rows

Technical details
Node id
n-2ea32e7d80305e8d · 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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    2,     16;
   3:    6,     80,     144;
   4:   16,    320,    1152,     1024;
   5:   44,   1152,    6192,    11264,      6400;
   6:  120,   3904,   27936,    76800,     89600,     36864;
   7:  328,  12736,  113760,   418816,    742400,    626688,    200704;
   8:  896,  40448,  433152,  1998848,   4761600,   6119424,   4014080,   1048576;
   9: 2448, 125952, 1572480,  8724480,  26118400,  45195264,  45158400,  24117248,   5308416;
  10: 6688, 386304, 5508864, 35708928, 128780800, 280498176, 377724928, 307232768, 138018816, 26214400;
%F Formula
The row polynomials satisfy (9+(54*n-60)*x+(78*n^2-126*n+16)*x^2+(36*n^3-44*n^2-120*n+128)*x^3+(4*n^4+8*n^3-116*n^2+104*n+32)*x^4-(16*n^3-48*n+32)*x^5)*P(n,x) = (18+(108*n+24)*x+(156*n^2+276*n-88)*x^2+(72*n^3+440*n^2-80*n-128)*x^3+(8*n^4+192*n^3+120*n^2-576*n+64)*x^4+(16*n^4+64*n^3-368*n^2-352*n+128)*x^5-(64*n^3+192*n^2)*x^6)*P(n-2,x) + (18+(108*n+24)*x+(156*n^2+192*n-88)*x^2+(72*n^3+368*n^2-352*n-128)*x^3+(8*n^4+176*n^3-72*n^2-752*n+64)*x^4+(16*n^4+32*n^3-432*n^2-288*n+128)*x^5-(64*n^3+128*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 16*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((18 + (108*n + 24)*x + (156*n^2 + 276*n - 88)*x^2 + (72*n^3 + 440*n^2 - 80*n - 128)*x^3 + (8*n^4 + 192*n^3 + 120*n^2 - 576*n + 64)*x^4 + (16*n^4 + 64*n^3 - 368*n^2 - 352*n + 128)*x^5 - (64*n^3 + 192*n^2)*x^6)*P[n - 2, x] + (18 + (108*n + 24)*x + (156*n^2 + 192*n - 88)*x^2 + (72*n^3 + 368*n^2 - 352*n - 128)*x^3 + (8*n^4 + 176*n^3 - 72*n^2 - 752*n + 64)*x^4 + (16*n^4 + 32*n^3 - 432*n^2 - 288*n + 128)*x^5 - (64*n^3 + 128*n^2)*x^6)*P[n - 1, x])/(9 + (54*n - 60)*x + (78*n^2 - 126*n + 16)*x^2 + (36*n^3 - 44*n^2 - 120*n + 128)*x^3 + (4*n^4 + 8*n^3 - 116*n^2 + 104*n + 32)*x^4 - (16*n^3 - 48*n + 32)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 16*x; for(n = 3, N, my(i = n); v[i] = ((18 + (108*n + 24)*x + (156*n^2 + 276*n - 88)*x^2 + (72*n^3 + 440*n^2 - 80*n - 128)*x^3 + (8*n^4 + 192*n^3 + 120*n^2 - 576*n + 64)*x^4 + (16*n^4 + 64*n^3 - 368*n^2 - 352*n + 128)*x^5 - (64*n^3 + 192*n^2)*x^6)*v[i - 2] + (18 + (108*n + 24)*x + (156*n^2 + 192*n - 88)*x^2 + (72*n^3 + 368*n^2 - 352*n - 128)*x^3 + (8*n^4 + 176*n^3 - 72*n^2 - 752*n + 64)*x^4 + (16*n^4 + 32*n^3 - 432*n^2 - 288*n + 128)*x^5 - (64*n^3 + 128*n^2)*x^6)*v[i - 1])/(9 + (54*n - 60)*x + (78*n^2 - 126*n + 16)*x^2 + (36*n^3 - 44*n^2 - 120*n + 128)*x^3 + (4*n^4 + 8*n^3 - 116*n^2 + 104*n + 32)*x^4 - (16*n^3 - 48*n + 32)*x^5)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.