Real-rooted atlas

A027467 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + n x\right)\,P_{n}(x) = \left(64 + 32\,x + 4\,x^{2} + 16\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116824
225625648560
340966144230425612800
46553613107273728163841280288000
510485762621440196608065536010240061446400000
616777216503316484718592020971520491520058982428672140800000
726843545693952409610569646085872025601835008003303014432112641310723072000000
84294967296171798691842254857830415032385536587202560014092861442055208961677721658982466560000000
96871947673630923764531246385646796836077725286416911433728050734301184986500300812079595528493465626214401433600000000
101099511627776549755813888092771293593608246337208320450971566080016234976378883946001203206442450944067947724804194304001153433630720000000000
1117592186044416967570232442881814194185830401814194185830401133871366144004762259737804813889924235264283467841536039862665216036909875200203004313650331648655360000000000
1228147497671065616888498602639363483252836794368387028092977152027212912787456001306219813797888444477575528448108851651149824191340793036802362232012800194884141056966367641621810380813926400000000000
134503599627370496292733975779082246586514455029350480501843339247616628920651087872003396171515874508813207333672845312377352390652723279597769903308812283606466560013511967113216100502234726445365592064939524096294912000000000000
147205759403792793650440315826549555212294826982721454081639310264362860544140878225843683328084526935506209996836980534283966873612075276500887142429716500763901952550305569701888075667015834009675041668595712508094631116821045339750440265318406225920000000000000
Technical details
Node id
n-f5b145ff37c5f0de · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-af50e1602aec03e4

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:          16,            8;
  2:         256,          256,           48;
  3:        4096,         6144,         2304,          256;
  4:       65536,       131072,        73728,        16384,         1280;
  5:     1048576,      2621440,      1966080,       655360,       102400,        6144;
  6:    16777216,     50331648,     47185920,     20971520,      4915200,      589824,      28672;
  7:   268435456,    939524096,   1056964608,    587202560,    183500800,    33030144,    3211264,     131072;
  8:  4294967296,  17179869184,  22548578304,  15032385536,   5872025600,  1409286144,  205520896,   16777216,   589824;
  9: 68719476736, 309237645312, 463856467968, 360777252864, 169114337280, 50734301184, 9865003008, 1207959552, 84934656, 2621440;
%F Formula
The row polynomials satisfy (4+n*x)*P(n,x) = (64+(16*n+32)*x+(4*n+4)*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 + (16*n + 32)*x + (4*n + 4)*x^2)*P[n - 1, x])/(4 + 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 + (16*n + 32)*x + (4*n + 4)*x^2)*v[i - 1])/(4 + 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.