Real-rooted atlas

A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x + n + 4\,n x - n x^{2} - n^{2} x + n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n}(x) = \left(8\,n + 14\,n x + 7\,n x^{2} + n x^{3} + 6\,n^{2} + 7\,n^{2} x + 4\,n^{2} x^{2} + n^{2} x^{3} + 2\,n^{3} + 3\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
214620
3666012138
425636015620792
588016801140320304050
627846720624027605704219116
7828824192285601736056709245685050
82355280640115584896004032010416140072361584
96451225344042739240320023184082656176402016901482786
101715207603201474560163968011491205241601545602808027901105904900
1144492821964804815360616704051004802838528107184026928042570374013222950378
1211304966150144150374402179584020782080136604166297984202752044352062040488415687392520
132822144167731204524748873216000790732805996390432672640128494083603600697840875166240182327013362
1469386244472832013194854423575552028444416024446822415375360071495424245044806086080105705612012078262101205308188
Technical details
Node id
n-4f31a406498f493f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ab77a325614cef16

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     2;
  2:    14,      6;
  3:    66,     60,     12;
  4:   256,    360,    156,     20;
  5:   880,   1680,   1140,    320,     30;
  6:  2784,   6720,   6240,   2760,    570,    42;
  7:  8288,  24192,  28560,  17360,   5670,   924,    56;
  8: 23552,  80640, 115584,  89600,  40320, 10416,  1400,   72;
  9: 64512, 253440, 427392, 403200, 231840, 82656, 17640, 2016, 90;
%F Formula
The row polynomials satisfy (n^3+n-2+(n^3-n^2+4*n-4)*x+(n^2-n)*x^2)*P(n,x) = (2*n^3+6*n^2+8*n+(3*n^3+7*n^2+14*n)*x+(n^3+4*n^2+7*n)*x^2+(n^2+n)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 6*n^2 + 8*n + (3*n^3 + 7*n^2 + 14*n)*x + (n^3 + 4*n^2 + 7*n)*x^2 + (n^2 + n)*x^3)*P[n - 1, x])/(n^3 + n - 2 + (n^3 - n^2 + 4*n - 4)*x + (n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^3 + 6*n^2 + 8*n + (3*n^3 + 7*n^2 + 14*n)*x + (n^3 + 4*n^2 + 7*n)*x^2 + (n^2 + n)*x^3)*v[i - 1])/(n^3 + n - 2 + (n^3 - n^2 + 4*n - 4)*x + (n^2 - n)*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

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.