Real-rooted atlas

A147716 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + x + n\right)\,P_{n}(x) = \left(196 + 28\,x + x^{2} + 14\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115116
2224301255
333326724514050
44939213328134460164125
57299042469603332022407511012500
610756480437942474088066640336090115946875
71581202567529536015327984172872011662047041051250593750
823190970881264962048301181440408746243457440186592627212013929765625
9339431482882087187379256923292169035443209196790462233922798888064135161509375000
1049586512281633943148288010435936896018974430720225886080018393580810372320399840100801501961083984375
117231366374400545451635097618668731558403826510195205217968448049694937603372156481629936054978012320165114992910156250
12105288694411264867763964928003272709810585674674926233601147953058560125231242752993898752057808396824449040733040147841801233543408203125
131530735634132992136875302734643256404657720320014181742512537624269351025920298467795225627133435929618458119680939386448353152809529521747219513632897460937500
1422224013651116032214302988778618889581271191425024263221736028160049636098793881667954182872576696424855526454266871859232301709440146126780849441392121284820384210156439656982421875
Technical details
Node id
n-7f7b0243d6933296 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-21c5d6e71fa08c4b

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:          15,           1;
  2:         224,          30,          1;
  3:        3332,         672,         45,         1;
  4:       49392,       13328,       1344,        60,        1;
  5:      729904,      246960,      33320,      2240,       75,       1;
  6:    10756480,     4379424,     740880,     66640,     3360,      90,      1;
  7:   158120256,    75295360,   15327984,   1728720,   116620,    4704,    105,    1;
  8:  2319097088,  1264962048,  301181440,  40874624,  3457440,  186592,   6272,  120,   1;
  9: 33943148288, 20871873792, 5692329216, 903544320, 91967904, 6223392, 279888, 8064, 135, 1;
%F Formula
The row polynomials satisfy (n+13+x)*P(n,x) = (14*n+196+(n+28)*x+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[((14*n + 196 + (n + 28)*x + x^2)*P[n - 1, x])/(n + 13 + 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] = ((14*n + 196 + (n + 28)*x + x^2)*v[i - 1])/(n + 13 + 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

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.