Real-rooted atlas

A038279 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(81 + 36\,x + 4\,x^{2} + 18\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111213
2117444165
3121570213282057
412393972028083521625289
512465912393048600936088032307461
612400291495908743580194400280802112643704173
7122231431736040610471356347004068040078624492812844289025
81195742251955702881388832485584723213880160217728020966411264256526153617
9116226146721523360501760132592833299488251312544499685766531840539136253445126216407549
10112351941812324522934021523360500117342172804166497440100525017616656192018662400134784056320102473096378421
11108090316431247174271982255697522740157837977000645381950401833258873636859173125234803205132160032947201239042048855935011833
12103557496709725941675943442966091263784204558018192094702786200030978333619273330354944126374307841570440960136857600790732827033640969985908471385
1398850337768352692494914452233724178726472257061242861281329627118248049245448824001342394456832272369889792410716500484536829440355829760186900485857288192116121849938677
14940490356481732767809457513803769492880233083147590014470721799428700028967445911862188822981209451200536957782732895329461427212777846681612703122432905748480436101121261568163841346385773613309
Technical details
Node id
n-47925298eb66eb09 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-eee8dca24d06ae1a

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:         11,          2;
  2:        117,         44,          4;
  3:       1215,        702,        132,         8;
  4:      12393,       9720,       2808,       352,        16;
  5:     124659,     123930,      48600,      9360,       880,       32;
  6:    1240029,    1495908,     743580,    194400,     28080,     2112,      64;
  7:   12223143,   17360406,   10471356,   3470040,    680400,    78624,    4928,    128;
  8:  119574225,  195570288,  138883248,  55847232,  13880160,  2177280,  209664,  11264,   256;
  9: 1162261467, 2152336050, 1760132592, 833299488, 251312544, 49968576, 6531840, 539136, 25344, 512;
%F Formula
The row polynomials satisfy (2*n+7+2*x)*P(n,x) = (18*n+81+(4*n+36)*x+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[((18*n + 81 + (4*n + 36)*x + 4*x^2)*P[n - 1, x])/(2*n + 7 + 2*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] = ((18*n + 81 + (4*n + 36)*x + 4*x^2)*v[i - 1])/(2*n + 7 + 2*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.