Real-rooted atlas

A038231 → D → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 - x + n + n x + n^{2}\right)\,P_{n}(x) = \left(12\,n + 7\,n x + n x^{2} + 4\,n^{2} + n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
210212
372303105
4448288604800
52560224072010055625
6138241536067201440150637500
77168096768537601568025202107240625
836044857344038707214336031360403228081500000
9176947232440322580480116121632256056448604836099140625
1085196801769472016220160860160029030406451209408086404501054687500
114037017693716480973209605947392023654400638668811827201478401188055011322265625
121887436804844421125622988803892838401784217605677056012773376202752022176015840660121875000000
138724152322453667840314887372824366284801265172480463896576123002880237219843294720320320205927801310791015625
143992977408122138132481717567488014694744064852819968035424829441082425344246005760415134725125120448448262089101461523437500
Technical details
Node id
n-0eb9fb88eb406743 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-76caa1d365acf455

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       1;
  2:      10,       2;
  3:      72,      30,       3;
  4:     448,     288,      60,       4;
  5:    2560,    2240,     720,     100,      5;
  6:   13824,   15360,    6720,    1440,    150,     6;
  7:   71680,   96768,   53760,   15680,   2520,   210,    7;
  8:  360448,  573440,  387072,  143360,  31360,  4032,  280,   8;
  9: 1769472, 3244032, 2580480, 1161216, 322560, 56448, 6048, 360, 9;
%F Formula
The row polynomials satisfy (n^2+n-2+(n-1)*x)*P(n,x) = (4*n^2+12*n+(n^2+7*n)*x+n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 12*n + (n^2 + 7*n)*x + n*x^2)*P[n - 1, x])/(n^2 + n - 2 + (n - 1)*x)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 12*n + (n^2 + 7*n)*x + n*x^2)*v[i - 1])/(n^2 + n - 2 + (n - 1)*x)); 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.