Real-rooted atlas

A038220 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
26072132
345010806482178
430001080012960518431944
5187509000016200012960038880439230
61125006750001620000194400011664002799365797836
765625047250001417500022680000204120009797760195955274405562
83750000315000001134000002268000002721600001959552007838208013436928935384208
92109375020250000085050000020412000003061800000293932800017635968006046617609069926411575379574
101171875001265625000607500000017010000000306180000003674160000029393280000151165440004534963200604661760141476861460
116445312507734375000417656250001336500000002806650000004041576000004041576000002771366400001247114880003325639680039907676161711870023666
123515625000464062500002784375000001002375000000240570000000040415760000004849891200000415704960000024942297600009976919040002394460569602612138803220542440283992
13190429687502742187500001809843750000723937500000019546312500000375289200000005254048800000054041644800000405312336000002161665792000077819968512001697890222080169789022208244797413384238
1410253906250015996093750001151718750000050675625000000152026875000000328378050000000525404880000000630485856000000567437270400000378291513600000181579926528000594261577728001188523155456010970982973442899907820090204
Technical details
Node id
n-4db9b5235c5a3916 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-da2debf48cb6165b

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:        6;
  2:       60,        72;
  3:      450,      1080,       648;
  4:     3000,     10800,     12960,       5184;
  5:    18750,     90000,    162000,     129600,      38880;
  6:   112500,    675000,   1620000,    1944000,    1166400,     279936;
  7:   656250,   4725000,  14175000,   22680000,   20412000,    9797760,    1959552;
  8:  3750000,  31500000, 113400000,  226800000,  272160000,  195955200,   78382080,  13436928;
  9: 21093750, 202500000, 850500000, 2041200000, 3061800000, 2939328000, 1763596800, 604661760, 90699264;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+6*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 6*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] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 6*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.