Real-rooted atlas

A013609 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(3 + 12\,x + 12\,x^{2} + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19615
24510836189
31898109722162187
4729453697207776129624057
52673218706804097200583207776255879
6947796228393660816480874800419904466562657205
7328053980342020788551124085730407348320293932827993627103491
81115371574640955281632332608661348808230118458786560201553921679616272629233
9373977602299842515280171950688436490208714256704740710656453496320136048896100776962711943423
10124002922438620180689940850305600257926032052378824967142567040634894848034012224009069926406046617626732013741
1140743818184191474047446039751786801403004240034046236224576167074566734420352052378824960249422976005986151424362797056261508830075
1213286025293355432294630890417771387040715532162402020326105604085548346885926289909766060978316804190305996801795845427203918208204821767823362541865828329
13430467211036309950114408618487660403150434654204728011162301733442626423937280455246815795257781326620165252847874560326843867750412734176665602546835333121306069401624571369673847
14138706101361592456443525017900320344131744160868466158458219063943041562722242681631517087247360478009156584965392923817881644123922146304249589862645768913923665920164564744601678364164096236393522034597
Technical details
Node id
n-7f7ad0475cdb9085 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      9,       6;
  2:     45,     108,       36;
  3:    189,     810,      972,       216;
  4:    729,    4536,     9720,      7776,      1296;
  5:   2673,   21870,    68040,     97200,     58320,      7776;
  6:   9477,   96228,   393660,    816480,    874800,    419904,     46656;
  7:  32805,  398034,  2020788,   5511240,   8573040,   7348320,   2939328,    279936;
  8: 111537, 1574640,  9552816,  32332608,  66134880,  82301184,  58786560,  20155392,   1679616;
  9: 373977, 6022998, 42515280, 171950688, 436490208, 714256704, 740710656, 453496320, 136048896, 10077696;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (6*n+3+(12*n+12)*x+12*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 3 + (12*n + 12)*x + 12*x^2)*P[n - 1, x])/(2*n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 3 + (12*n + 12)*x + 12*x^2)*v[i - 1])/(2*n - 1 + 2*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.