Real-rooted atlas

A331333 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
28816
3367224132
4192576384641216
5120048004800160016012560
686404320057600288005760384144384
770560423360705600470400141120188168961830752
8645120451584090316807526400301056060211257344204825391104
9653184052254720121927680121927680609638401625702423224321658884608382355712
1072576000653184000174182400020321280001219276800406425600774144008294400460800102406211594240
1187816960087816960002634508800035126784000245887488009835499520234178560033454080027878400123904022528108261452288
121149603840012645642240042152140800063228211200050582568960023605198848067443425280120434688001338163200892108803244032491522014551220224
1316190254080019428304896007123711795200118728529920001068556769280056989694361601899656478720407069245440565373952005025546240274120704830668810649639854406146048
14244099215360031732897996800126931591987200232707918643200232707918643200139624751185920531903814041601329759535104022162658918402462517657601790921932881405542420873216229376835115309400064
Technical details
Node id
n-4e47c45c71c4060b · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-269631c8dfab89fe

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:       8,        8;
  3:      36,       72,        24;
  4:     192,      576,       384,        64;
  5:    1200,     4800,      4800,      1600,      160;
  6:    8640,    43200,     57600,     28800,     5760,      384;
  7:   70560,   423360,    705600,    470400,   141120,    18816,     896;
  8:  645120,  4515840,   9031680,   7526400,  3010560,   602112,   57344,   2048;
  9: 6531840, 52254720, 121927680, 121927680, 60963840, 16257024, 2322432, 165888, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n^2+2*n*x)*P(n-1,x) + n*x*d/dx 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[((n^2 + 2*n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], 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] = ((n^2 + 2*n*x)*v[i - 1] + n*x*deriv(v[i - 1], x))/(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.