Real-rooted atlas

A081577 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(36\,n x + 108\,n x^{2} + 54\,n x^{3} + 54\,n^{2} x^{2} + 81\,n^{2} x^{3} + 27\,n^{3} x^{3}\right)\,P_{n}(x) = \left(48 + 360\,x + 900\,x^{2} + 864\,x^{3} + 324\,x^{4} - 48\,n - 180\,n x + 72\,n x^{2} + 729\,n x^{3} + 486\,n x^{4} - 108\,n^{2} x - 378\,n^{2} x^{2} - 162\,n^{2} x^{3} + 162\,n^{2} x^{4} - 54\,n^{3} x^{2} - 135\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(16 + 168\,x + 660\,x^{2} + 1170\,x^{3} + 864\,x^{4} + 162\,x^{5} + 48\,n x + 396\,n x^{2} + 1116\,n x^{3} + 1161\,n x^{4} + 243\,n x^{5} + 36\,n^{2} x^{2} + 216\,n^{2} x^{3} + 351\,n^{2} x^{4} + 81\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1268
26362769
316144270108538
444528156615124054055
512018007344116647290145829676
63285904306726825670470320765103212809
789618816118800341280498960373248132678174961502174
824485875243675215396482930580311428818064625248805904910472859
96688180576154483264592641519236021117672173910248188128200766619683072285040
10182725481605304528256711687201548012425659213410684089439552353113027479540649539494800973
1149920164736017790624978376323191270406593892488857175047743029764317662881464415202728063821257643363476514
1213638449098245853686436061632013417099203234660480521730720056838205444151144700198247176058848233497785144690873322728490207
13372608145317121895754241293380352540776736014915969856281430858243687764889633575261400209734174808740196784230432817634543665022320522152803293044
14101798442755328605805696453458995221053753280654295345921415693039042173893396482382806658961855036144801009539495363726039139288293607741205308188717445351022731135185
Technical details
Node id
n-4f216711705dda85 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-dc06298ca97775b4

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,      6;
  2:    6,     36,      27;
  3:   16,    144,     270,     108;
  4:   44,    528,    1566,    1512,      405;
  5:  120,   1800,    7344,   11664,     7290,     1458;
  6:  328,   5904,   30672,   68256,    70470,    32076,     5103;
  7:  896,  18816,  118800,  341280,   498960,   373248,   132678,   17496;
  8: 2448,  58752,  436752, 1539648,  2930580,  3114288,  1806462,  524880,   59049;
  9: 6688, 180576, 1544832, 6459264, 15192360, 21117672, 17391024, 8188128, 2007666, 196830;
%F Formula
The row polynomials satisfy (36*n*x+(54*n^2+108*n)*x^2+(27*n^3+81*n^2+54*n)*x^3)*P(n,x) = -(48*n-48+(108*n^2+180*n-360)*x+(54*n^3+378*n^2-72*n-900)*x^2+(135*n^3+162*n^2-729*n-864)*x^3-(162*n^2+486*n+324)*x^4)*P(n-1,x) + (16+(48*n+168)*x+(36*n^2+396*n+660)*x^2+(216*n^2+1116*n+1170)*x^3+(351*n^2+1161*n+864)*x^4+(81*n^2+243*n+162)*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[(-(48*n - 48 + (108*n^2 + 180*n - 360)*x + (54*n^3 + 378*n^2 - 72*n - 900)*x^2 + (135*n^3 + 162*n^2 - 729*n - 864)*x^3 - (162*n^2 + 486*n + 324)*x^4)*P[n - 1, x] + (16 + (48*n + 168)*x + (36*n^2 + 396*n + 660)*x^2 + (216*n^2 + 1116*n + 1170)*x^3 + (351*n^2 + 1161*n + 864)*x^4 + (81*n^2 + 243*n + 162)*x^5)*D[P[n - 1, x], x])/(36*n*x + (54*n^2 + 108*n)*x^2 + (27*n^3 + 81*n^2 + 54*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] = (-(48*n - 48 + (108*n^2 + 180*n - 360)*x + (54*n^3 + 378*n^2 - 72*n - 900)*x^2 + (135*n^3 + 162*n^2 - 729*n - 864)*x^3 - (162*n^2 + 486*n + 324)*x^4)*v[i - 1] + (16 + (48*n + 168)*x + (36*n^2 + 396*n + 660)*x^2 + (216*n^2 + 1116*n + 1170)*x^3 + (351*n^2 + 1161*n + 864)*x^4 + (81*n^2 + 243*n + 162)*x^5)*deriv(v[i - 1], x))/(36*n*x + (54*n^2 + 108*n)*x^2 + (27*n^3 + 81*n^2 + 54*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.