Real-rooted atlas

A352880 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-48 + 12\,x + 3\,x^{2} - 112\,n + 28\,n x + 7\,n x^{2} - 80\,n^{2} + 20\,n^{2} x + 5\,n^{2} x^{2} - 16\,n^{3} + 4\,n^{3} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(-192 + 48\,x + 12\,x^{2} - 256\,n - 1280\,n x + 384\,n x^{2} + 160\,n x^{3} + 6\,n x^{4} - 112\,n^{2} - 164\,n^{2} x + 599\,n^{2} x^{2} + 224\,n^{2} x^{3} + 29\,n^{2} x^{4} - 16\,n^{3} + 4\,n^{3} x - 559\,n^{3} x^{2} + 76\,n^{3} x^{3} + 9\,n^{3} x^{4} - 16\,n^{4} x^{2} - 196\,n^{4} x^{3} - 28\,n^{4} x^{4} - 168\,n^{5} x^{3} - 16\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(-624\,x + 1188\,x^{2} - 456\,x^{3} - 228\,x^{4} - 12\,x^{5} - 256\,n x + 760\,n x^{2} - 484\,n x^{3} - 380\,n x^{4} - 58\,n x^{5} - 32\,n^{2} x - 40\,n^{2} x^{2} + 1166\,n^{2} x^{3} - 136\,n^{2} x^{4} - 18\,n^{2} x^{5} + 200\,n^{3} x^{3} + 408\,n^{3} x^{4} + 56\,n^{3} x^{5} + 336\,n^{4} x^{4} + 32\,n^{4} x^{5}\right)\,P_{n-1}'(x) + \left(-176\,x^{2} + 388\,x^{3} - 122\,x^{4} - 84\,x^{5} - 6\,x^{6} - 48\,n x^{2} + 468\,n x^{3} - 145\,n x^{4} - 240\,n x^{5} - 35\,n x^{6} + 16\,n^{2} x^{3} + 348\,n^{2} x^{4} - 320\,n^{2} x^{5} - 44\,n^{2} x^{6} + 168\,n^{3} x^{4} - 152\,n^{3} x^{5} - 16\,n^{3} x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
088
1184866
232256144432
35080013503202520
47219206480460860013680
5983920220503136012250100870686
6128716860480143360112000276481568352352
7162120961428845080326615003265925556623041709136

All 15 rows

Technical details
Node id
n-28612eb32b3440a9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2751883146f8224a

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:   8;
  1:  18,    48;
  2:  32,   256,    144;
  3:  50,   800,   1350,     320;
  4:  72,  1920,   6480,    4608,      600;
  5:  98,  3920,  22050,   31360,    12250,     1008;
  6: 128,  7168,  60480,  143360,   112000,    27648,    1568;
  7: 162, 12096, 142884,  508032,   661500,   326592,   55566,    2304;
  8: 200, 19200, 302400, 1505280,  2940000,  2419200,  823200,  102400,   3240;
  9: 242, 29040, 588060, 3902976, 10672200, 13172544, 7470540, 1858560, 176418, 4400;
%F Formula
The row polynomials satisfy -(16*n^3+80*n^2+112*n+48-(4*n^3+20*n^2+28*n+12)*x-(n^3+5*n^2+7*n+3)*x^2)*P(n,x) = -(16*n^3+112*n^2+256*n+192-(4*n^3-164*n^2-1280*n+48)*x+(16*n^4+559*n^3-599*n^2-384*n-12)*x^2+(168*n^5+196*n^4-76*n^3-224*n^2-160*n)*x^3+(16*n^5+28*n^4-9*n^3-29*n^2-6*n)*x^4)*P(n-1,x) - ((32*n^2+256*n+624)*x+(40*n^2-760*n-1188)*x^2-(200*n^3+1166*n^2-484*n-456)*x^3-(336*n^4+408*n^3-136*n^2-380*n-228)*x^4-(32*n^4+56*n^3-18*n^2-58*n-12)*x^5)*d/dx P(n-1,x) - ((48*n+176)*x^2-(16*n^2+468*n+388)*x^3-(168*n^3+348*n^2-145*n-122)*x^4+(152*n^3+320*n^2+240*n+84)*x^5+(16*n^3+44*n^2+35*n+6)*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 8.
%t Mathematica
P[0, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n^3 + 112*n^2 + 256*n + 192 - (4*n^3 - 164*n^2 - 1280*n + 48)*x + (16*n^4 + 559*n^3 - 599*n^2 - 384*n - 12)*x^2 + (168*n^5 + 196*n^4 - 76*n^3 - 224*n^2 - 160*n)*x^3 + (16*n^5 + 28*n^4 - 9*n^3 - 29*n^2 - 6*n)*x^4)*P[n - 1, x] - ((32*n^2 + 256*n + 624)*x + (40*n^2 - 760*n - 1188)*x^2 - (200*n^3 + 1166*n^2 - 484*n - 456)*x^3 - (336*n^4 + 408*n^3 - 136*n^2 - 380*n - 228)*x^4 - (32*n^4 + 56*n^3 - 18*n^2 - 58*n - 12)*x^5)*D[P[n - 1, x], x] - ((48*n + 176)*x^2 - (16*n^2 + 468*n + 388)*x^3 - (168*n^3 + 348*n^2 - 145*n - 122)*x^4 + (152*n^3 + 320*n^2 + 240*n + 84)*x^5 + (16*n^3 + 44*n^2 + 35*n + 6)*x^6)*D[P[n - 1, x], {x, 2}])/(-16*n^3 - 80*n^2 - 112*n - 48 + (4*n^3 + 20*n^2 + 28*n + 12)*x + (n^3 + 5*n^2 + 7*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8; for(n = 1, N, my(i = n + 1); v[i] = (-(16*n^3 + 112*n^2 + 256*n + 192 - (4*n^3 - 164*n^2 - 1280*n + 48)*x + (16*n^4 + 559*n^3 - 599*n^2 - 384*n - 12)*x^2 + (168*n^5 + 196*n^4 - 76*n^3 - 224*n^2 - 160*n)*x^3 + (16*n^5 + 28*n^4 - 9*n^3 - 29*n^2 - 6*n)*x^4)*v[i - 1] - ((32*n^2 + 256*n + 624)*x + (40*n^2 - 760*n - 1188)*x^2 - (200*n^3 + 1166*n^2 - 484*n - 456)*x^3 - (336*n^4 + 408*n^3 - 136*n^2 - 380*n - 228)*x^4 - (32*n^4 + 56*n^3 - 18*n^2 - 58*n - 12)*x^5)*deriv(v[i - 1], x) - ((48*n + 176)*x^2 - (16*n^2 + 468*n + 388)*x^3 - (168*n^3 + 348*n^2 - 145*n - 122)*x^4 + (152*n^3 + 320*n^2 + 240*n + 84)*x^5 + (16*n^3 + 44*n^2 + 35*n + 6)*x^6)*deriv(deriv(v[i - 1], x), x))/(-16*n^3 - 80*n^2 - 112*n - 48 + (4*n^3 + 20*n^2 + 28*n + 12)*x + (n^3 + 5*n^2 + 7*n + 3)*x^2)); 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.