Real-rooted atlas

A013622 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 + 5\,n x\right)\,P_{n}(x) = \left(64 + 80\,x + 25\,x^{2} + 40\,n x + 25\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181018
26416075299
3512192018005004732
44096204802880016000312572501
532768204800384000320000125000187501085318
6262144196608046080005120000300000090000010937515965599
7209715218350080516096007168000056000000252000006125000625000231686832
8167772161677721605505024009175040008960000005376000001960000004000000035156253325671401
91342177281509949440566231040011010048000129024000009676800000470400000014400000002531250001953125047312381818
10107374182413421772800566231040001258291200001720320000001548288000009408000000038400000000101250000001562500000107421875668083460499
11858993459211811160064055364812800013841203200002162688000000227082240000016558080000008448000000002970000000006875000000094531250005859375009374377445732
126871947673610307921510405315022028800147639500800002595225600000031142707200000264929280000001622016000000071280000000002200000000000453750000000562500000003173828125130827708764701
1354975581388889335319756805025111736320015354508083200029989273600000040485519360000039360921600000028114944000000014826240000000057200000000000157300000000002925000000000330078125000170898437501817250639553518
1443980465111047696581394432046901042872320015633680957440003358798643200000503819796480000055105290240000004498391040000000276756480000000012812800000000004404400000000001092000000000001848437500000019140625000009155273437525138633847156999
Technical details
Node id
n-ba39776209ab0396 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3393477035eb5c26

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:         8,         10;
  2:        64,        160,         75;
  3:       512,       1920,       1800,         500;
  4:      4096,      20480,      28800,       16000,        3125;
  5:     32768,     204800,     384000,      320000,      125000,      18750;
  6:    262144,    1966080,    4608000,     5120000,     3000000,     900000,     109375;
  7:   2097152,   18350080,   51609600,    71680000,    56000000,   25200000,    6125000,     625000;
  8:  16777216,  167772160,  550502400,   917504000,   896000000,  537600000,  196000000,   40000000,   3515625;
  9: 134217728, 1509949440, 5662310400, 11010048000, 12902400000, 9676800000, 4704000000, 1440000000, 253125000, 19531250;
%F Formula
The row polynomials satisfy (8+5*n*x)*P(n,x) = (64+(40*n+80)*x+(25*n+25)*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[((64 + (40*n + 80)*x + (25*n + 25)*x^2)*P[n - 1, x])/(8 + 5*n*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] = ((64 + (40*n + 80)*x + (25*n + 25)*x^2)*v[i - 1])/(8 + 5*n*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.