Real-rooted atlas

A038208 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(8 + 24\,x + 18\,x^{2} + 12\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
26412036220
3352115210802162800
417928448138248640129634000
5870453760126720138240648007776400000
64096031334496768015206401244160466560466564600000
718841617203206580224135475201596672010450944326592027993652000000
885196890439684128768010528358416257024015328051283607552223948801679616580000000
9380108846006272244187136743178240142132838417557585921379524608644972544151165440100776966400000000
10167772162280652801380188160488374272011147673600170559406081755758592011824496640483729408010077696006046617670000000000
117340032011072962567526154240303641395208058175488014714929152018761534668816554295296097552097280354734899206651279360362797056760000000000
1231876710452848230403986266521618062770176054655451136011603772702721765791498240192975785164814898865766407804167782402554091274244353564672021767823368200000000000
1313757317122486383411220610809856010364292956163522240184320852625037721615084904513536196759624089601881513905356812912350330880608725087027218110829035522829817036801306069401688000000000000
1459055800321155614638081044281032704577102675968021765015207936591736350965761193675052810241810188541624322065976052940801756079644999681084637427793924648446119116812677580324864182849716224078364164096940000000000000
Technical details
Node id
n-623ef47addb9e7aa · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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:      10,        6;
  2:      64,      120,        36;
  3:     352,     1152,      1080,       216;
  4:    1792,     8448,     13824,      8640,       1296;
  5:    8704,    53760,    126720,    138240,      64800,       7776;
  6:   40960,   313344,    967680,   1520640,    1244160,     466560,      46656;
  7:  188416,  1720320,   6580224,  13547520,   15966720,   10450944,    3265920,    279936;
  8:  851968,  9043968,  41287680, 105283584,  162570240,  153280512,   83607552,  22394880,   1679616;
  9: 3801088, 46006272, 244187136, 743178240, 1421328384, 1755758592, 1379524608, 644972544, 151165440, 10077696;
%F Formula
The row polynomials satisfy (3*n-1+3*x)*P(n,x) = (12*n+8+(18*n+24)*x+18*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[((12*n + 8 + (18*n + 24)*x + 18*x^2)*P[n - 1, x])/(3*n - 1 + 3*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] = ((12*n + 8 + (18*n + 24)*x + 18*x^2)*v[i - 1])/(3*n - 1 + 3*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.