Real-rooted atlas

A099040 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 12\,x - 32\,x^{2} + 8\,x^{4} + 108\,n x + 30\,n x^{2} - 50\,n x^{3} - 22\,n x^{4} + 78\,n^{2} x^{2} + 32\,n^{2} x^{3} - 17\,n^{2} x^{4} - 6\,n^{2} x^{5} + 18\,n^{3} x^{3} + 6\,n^{3} x^{4} - 2\,n^{3} x^{5} + n^{4} x^{4}\right)\,P_{n}(x) = \left(72 + 264\,x + 344\,x^{2} + 152\,x^{3} - 48\,x^{4} - 64\,x^{5} - 16\,x^{6} + 216\,n x + 588\,n x^{2} + 508\,n x^{3} + 68\,n x^{4} - 104\,n x^{5} - 36\,n x^{6} + 156\,n^{2} x^{2} + 328\,n^{2} x^{3} + 186\,n^{2} x^{4} - 10\,n^{2} x^{5} - 24\,n^{2} x^{6} + 36\,n^{3} x^{3} + 56\,n^{3} x^{4} + 16\,n^{3} x^{5} - 4\,n^{3} x^{6} + 2\,n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(72 + 264\,x + 260\,x^{2} - 20\,x^{3} - 144\,x^{4} - 68\,x^{5} - 12\,x^{6} + 216\,n x + 504\,n x^{2} + 300\,n x^{3} - 84\,n x^{4} - 124\,n x^{5} - 28\,n x^{6} + 156\,n^{2} x^{2} + 292\,n^{2} x^{3} + 126\,n^{2} x^{4} - 30\,n^{2} x^{5} - 20\,n^{2} x^{6} + 36\,n^{3} x^{3} + 52\,n^{3} x^{4} + 12\,n^{3} x^{5} - 4\,n^{3} x^{6} + 2\,n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12810
26403682
316160288128592
444576154814084003976
51201952698496005600115225408
63286368284405235246400195843136156608
789620224108288249856297600191232627208192939008
8244862976393120109056016324001412352705600188416207365508608
966881931521377216446361680488008765568590195224002565391365120031747584
10182725853444693248174028803671920048237696409216642243788876723201484800123904180297216
114992017566721564416065316864157868800242938368248822784171991040792944642334720039649282949121011290112
1213638452295685121388823775027264767520011427655681372134848114574131266608179226439680068271104103219206922245612410880
133726081546291216515417684385177625581728005091374592701410931268698030084826262528241592320084081254419316736026304512160563230862376960
1410179844545689652585228829325127689791702400217011225603374320320037938012160312740352001892198400083045416962570747904531628032658309123686400168351334400
Technical details
Node id
n-3a17caa08883823f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b7c8d43fd9e53efa

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,      8;
  2:    6,     40,      36;
  3:   16,    160,     288,     128;
  4:   44,    576,    1548,    1408,     400;
  5:  120,   1952,    6984,    9600,    5600,    1152;
  6:  328,   6368,   28440,   52352,   46400,   19584,    3136;
  7:  896,  20224,  108288,  249856,  297600,  191232,   62720,    8192;
  8: 2448,  62976,  393120, 1090560, 1632400, 1412352,  705600,  188416,  20736;
  9: 6688, 193152, 1377216, 4463616, 8048800, 8765568, 5901952, 2400256, 539136, 51200;
%F Formula
The row polynomials satisfy (36+(108*n-12)*x+(78*n^2+30*n-32)*x^2+(18*n^3+32*n^2-50*n)*x^3+(n^4+6*n^3-17*n^2-22*n+8)*x^4-(2*n^3+6*n^2)*x^5)*P(n,x) = (72+(216*n+264)*x+(156*n^2+588*n+344)*x^2+(36*n^3+328*n^2+508*n+152)*x^3+(2*n^4+56*n^3+186*n^2+68*n-48)*x^4+(2*n^4+16*n^3-10*n^2-104*n-64)*x^5-(4*n^3+24*n^2+36*n+16)*x^6)*P(n-2,x) + (72+(216*n+264)*x+(156*n^2+504*n+260)*x^2+(36*n^3+292*n^2+300*n-20)*x^3+(2*n^4+52*n^3+126*n^2-84*n-144)*x^4+(2*n^4+12*n^3-30*n^2-124*n-68)*x^5-(4*n^3+20*n^2+28*n+12)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((72 + (216*n + 264)*x + (156*n^2 + 588*n + 344)*x^2 + (36*n^3 + 328*n^2 + 508*n + 152)*x^3 + (2*n^4 + 56*n^3 + 186*n^2 + 68*n - 48)*x^4 + (2*n^4 + 16*n^3 - 10*n^2 - 104*n - 64)*x^5 - (4*n^3 + 24*n^2 + 36*n + 16)*x^6)*P[n - 2, x] + (72 + (216*n + 264)*x + (156*n^2 + 504*n + 260)*x^2 + (36*n^3 + 292*n^2 + 300*n - 20)*x^3 + (2*n^4 + 52*n^3 + 126*n^2 - 84*n - 144)*x^4 + (2*n^4 + 12*n^3 - 30*n^2 - 124*n - 68)*x^5 - (4*n^3 + 20*n^2 + 28*n + 12)*x^6)*P[n - 1, x])/(36 + (108*n - 12)*x + (78*n^2 + 30*n - 32)*x^2 + (18*n^3 + 32*n^2 - 50*n)*x^3 + (n^4 + 6*n^3 - 17*n^2 - 22*n + 8)*x^4 - (2*n^3 + 6*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((72 + (216*n + 264)*x + (156*n^2 + 588*n + 344)*x^2 + (36*n^3 + 328*n^2 + 508*n + 152)*x^3 + (2*n^4 + 56*n^3 + 186*n^2 + 68*n - 48)*x^4 + (2*n^4 + 16*n^3 - 10*n^2 - 104*n - 64)*x^5 - (4*n^3 + 24*n^2 + 36*n + 16)*x^6)*v[i - 2] + (72 + (216*n + 264)*x + (156*n^2 + 504*n + 260)*x^2 + (36*n^3 + 292*n^2 + 300*n - 20)*x^3 + (2*n^4 + 52*n^3 + 126*n^2 - 84*n - 144)*x^4 + (2*n^4 + 12*n^3 - 30*n^2 - 124*n - 68)*x^5 - (4*n^3 + 20*n^2 + 28*n + 12)*x^6)*v[i - 1])/(36 + (108*n - 12)*x + (78*n^2 + 30*n - 32)*x^2 + (18*n^3 + 32*n^2 - 50*n)*x^3 + (n^4 + 6*n^3 - 17*n^2 - 22*n + 8)*x^4 - (2*n^3 + 6*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.