Real-rooted atlas

A038255 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
228836
329416824486
427442352672645832
5240102744011760224016065610
6201684288120164640470406720384708588
7164708628235762016840768320164640188168967440174
813176688263533762258860810756480307328052684850176204876527504
9103766418237180384237180384135531648484041601106380815805441290244608774840978
10807072140207532836023718038401581202560677658240193616640368793604515840322560102407748409780
11621445547817755587080228286119601739322816086966140802981696256709927680115906560124185607884802252876709256822
12474558418321491469314722130670449601826288956801043593689604174374758411926785024243403776034771968033116160189235249152753145430616
13359873467226123385188763219389101091361846581056320118708782192054266871859218088957286444299487232791062272010045235208610201644728321064967343167948506
14271289229139610076457082328172739264268481809649435193612926067394240664769180275225324540200967235582914561550482053122461082624028126658562191687681043660822937671172243193212
Technical details
Node id
n-6890f01c78bc6b64 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8d10fbbacee008e8

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:         2;
  2:        28,         8;
  3:       294,       168,        24;
  4:      2744,      2352,       672,        64;
  5:     24010,     27440,     11760,      2240,      160;
  6:    201684,    288120,    164640,     47040,     6720,      384;
  7:   1647086,   2823576,   2016840,    768320,   164640,    18816,     896;
  8:  13176688,  26353376,  22588608,  10756480,  3073280,   526848,   50176,   2048;
  9: 103766418, 237180384, 237180384, 135531648, 48404160, 11063808, 1580544, 129024, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (7*n+2*n*x)*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[((7*n + 2*n*x)*P[n - 1, x])/(n - 1)]];
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] = ((7*n + 2*n*x)*v[i - 1])/(n - 1)); 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.