Real-rooted atlas

A038279 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 + n x\right)\,P_{n}(x) = \left(64 + 16\,x + x^{2} + 8\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18210
26432399
3512384724972
440964096115212859477
53276840960153602560200691854
62621443932161843204096048002887885735
720971523670016206438457344089600806439288503056
81677721633554432220200967340032143360017203212544512981310473
9134217728301989888226492416880803842064384030965763010561843264810774840978
10107374182426843545602264924160100663296027525120049545216602112049152025920800117360989291
118589934592236223201282214592512011072962560346030080072666316810597171210813440760320352009681269735688020
1268719476736206158430208212600881152118111600640415236096009965666304169554739220761804818247680112640046464115213659002251789
13549755813888178670639513620100446945281228360646656479828377600129553661952251909898243598712832379551744292864001610752599041352146213449802582
144398046511104153931627888641876041714892812506944765952537407782912016122233487363526738575365757940531270849658886560153604510105622364167571215681558462914051567
Technical details
Node id
n-05676cf5d9d8ddaf · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-713ec18753f9f2c0

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,         2;
  2:        64,        32,         3;
  3:       512,       384,        72,        4;
  4:      4096,      4096,      1152,      128,        5;
  5:     32768,     40960,     15360,     2560,      200,       6;
  6:    262144,    393216,    184320,    40960,     4800,     288,      7;
  7:   2097152,   3670016,   2064384,   573440,    89600,    8064,    392,     8;
  8:  16777216,  33554432,  22020096,  7340032,  1433600,  172032,  12544,   512,   9;
  9: 134217728, 301989888, 226492416, 88080384, 20643840, 3096576, 301056, 18432, 648, 10;
%F Formula
The row polynomials satisfy (8+n*x)*P(n,x) = (64+(8*n+16)*x+(n+1)*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 + (8*n + 16)*x + (n + 1)*x^2)*P[n - 1, x])/(8 + 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 + (8*n + 16)*x + (n + 1)*x^2)*v[i - 1])/(8 + 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.