Real-rooted atlas

A038327 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(12 + x + n\right)\,P_{n}(x) = \left(169 + 26\,x + x^{2} + 13\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114115
2195281224
327045854213332
43734910816117056149392
5514098186745270401950701729904
67054567308458856023554080292584110756480
79653618049381969107960581307215946404095981158120256
81317718857772289440197527876287894882614430151424546011212319097088
9179460758621185946971334753024805925836286477634847059742271367020126133943148288
1024390348557917946075862059297348565115843416001481459070129552696784329032448087751401495865122816
1133086038043762682938341369987034172410217423611405318569394003259209954237513276123251704461601072515417231366374400
12448040098509253970324565251216097630048214394813668964065227083421576456654560651841990840716561618487755594880128701681105288694411264
136057502131845065824521280620252580710967413286975639687559412831444241330169590416895916565608488012105636972661644126267045357733441521018211530735634132992
1481776278779908318480502984583084407716489643417512043317847928642441473890645793592804387572439571097275713313121697602118486470110292241963738634998425617745196122224013651116032
Technical details
Node id
n-72626dacb9dafe0c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0d4def4656af2b7d

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:          14,           1;
  2:         195,          28,          1;
  3:        2704,         585,         42,         1;
  4:       37349,       10816,       1170,        56,        1;
  5:      514098,      186745,      27040,      1950,       70,       1;
  6:     7054567,     3084588,     560235,     54080,     2925,      84,      1;
  7:    96536180,    49381969,   10796058,   1307215,    94640,    4095,     98,    1;
  8:  1317718857,   772289440,  197527876,  28789488,  2614430,  151424,   5460,  112,   1;
  9: 17946075862, 11859469713, 3475302480, 592583628, 64776348, 4705974, 227136, 7020, 126, 1;
%F Formula
The row polynomials satisfy (n+12+x)*P(n,x) = (13*n+169+(n+26)*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[((13*n + 169 + (n + 26)*x + x^2)*P[n - 1, x])/(n + 12 + 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] = ((13*n + 169 + (n + 26)*x + x^2)*v[i - 1])/(n + 12 + 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.