Real-rooted atlas

A081579 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n x + 6\,n x^{2} + 2\,n x^{3} + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(10 + 48\,x + 74\,x^{2} + 40\,x^{3} + 8\,x^{4} - 10\,n - 27\,n x - 2\,n x^{2} + 31\,n x^{3} + 12\,n x^{4} - 15\,n^{2} x - 37\,n^{2} x^{2} - 14\,n^{2} x^{3} + 4\,n^{2} x^{4} - 5\,n^{3} x^{2} - 9\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(5 + 35\,x + 91\,x^{2} + 105\,x^{3} + 48\,x^{4} + 4\,x^{5} + 10\,n x + 55\,n x^{2} + 102\,n x^{3} + 67\,n x^{4} + 6\,n x^{5} + 5\,n^{2} x^{2} + 20\,n^{2} x^{3} + 21\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
28321252
32414416832368
4806401200640802640
5256256067206400208019218208
683299843360046080268806144448123968
726883763215475227904024128097536170241024830976
887041392646766081513472174848010567683207684505623045511424
928160506880284160075980801103616090163204085760983040115200512036216320
10911361822720115814403604480063196160655319044035584014417920285696028672011264236196864
11294912648806446080000163676160336629760424747008334114816162365440474624007966720698368245761530548224
12954368229048321798225927179468801695682560252621619224323604481515061248601436160147947520214917121671168532489863548928
13308838480297984690536448306164531281695129601405270425616059990016122940293126262898688208617472044136857656426496394035211468863262728192
149994240279838720261611520012754616320379553382407409565696098217082880895955763205622865920024027791360686022656012700876801448345609175040245760404065239040
Technical details
Node id
n-248ee1970eb8522b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bbd6595a95368682

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,      4;
  2:     8,     32,      12;
  3:    24,    144,     168,      32;
  4:    80,    640,    1200,     640,       80;
  5:   256,   2560,    6720,    6400,     2080,     192;
  6:   832,   9984,   33600,   46080,    26880,    6144,     448;
  7:  2688,  37632,  154752,  279040,   241280,   97536,   17024,   1024;
  8:  8704, 139264,  676608, 1513472,  1748480, 1056768,  320768,  45056,   2304;
  9: 28160, 506880, 2841600, 7598080, 11036160, 9016320, 4085760, 983040, 115200, 5120;
%F Formula
The row polynomials satisfy (3*n*x+(3*n^2+6*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(10*n-10+(15*n^2+27*n-48)*x+(5*n^3+37*n^2+2*n-74)*x^2+(9*n^3+14*n^2-31*n-40)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (5+(10*n+35)*x+(5*n^2+55*n+91)*x^2+(20*n^2+102*n+105)*x^3+(21*n^2+67*n+48)*x^4+(2*n^2+6*n+4)*x^5)*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[(-(10*n - 10 + (15*n^2 + 27*n - 48)*x + (5*n^3 + 37*n^2 + 2*n - 74)*x^2 + (9*n^3 + 14*n^2 - 31*n - 40)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (5 + (10*n + 35)*x + (5*n^2 + 55*n + 91)*x^2 + (20*n^2 + 102*n + 105)*x^3 + (21*n^2 + 67*n + 48)*x^4 + (2*n^2 + 6*n + 4)*x^5)*D[P[n - 1, x], x])/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];
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] = (-(10*n - 10 + (15*n^2 + 27*n - 48)*x + (5*n^3 + 37*n^2 + 2*n - 74)*x^2 + (9*n^3 + 14*n^2 - 31*n - 40)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (5 + (10*n + 35)*x + (5*n^2 + 55*n + 91)*x^2 + (20*n^2 + 102*n + 105)*x^3 + (21*n^2 + 67*n + 48)*x^4 + (2*n^2 + 6*n + 4)*x^5)*deriv(v[i - 1], x))/(3*n*x + (3*n^2 + 6*n)*x^2 + (n^3 + 3*n^2 + 2*n)*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.