Real-rooted atlas

A038326 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(35 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1352459
2122516805763481
342875882006048013824205379
4150062541160004233600193536033177612117361
552521875180075000246960000169344000580608007962624714924299
61838265625756315000012965400000118540800006096384000167215104019110297642180533641
7643392968753088286250006353046000007260624000004978713600002048385024004682022912045864714242488651484819

All 15 rows

Technical details
Node id
n-d45b688c06f57565 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-d7d295a178a56855

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:             35,              24;
  2:           1225,            1680,              576;
  3:          42875,           88200,            60480,            13824;
  4:        1500625,         4116000,          4233600,          1935360,           331776;
  5:       52521875,       180075000,        246960000,        169344000,         58060800,          7962624;
  6:     1838265625,      7563150000,      12965400000,      11854080000,       6096384000,       1672151040,       191102976;
  7:    64339296875,    308828625000,     635304600000,     726062400000,     497871360000,     204838502400,     46820229120,      4586471424;
  8:  2251875390625,  12353145000000,   29647548000000,   40659494400000,   34850995200000,   19118260224000,   6554832076800,   1284211998720,   110075314176;
  9: 78815638671875, 486405084375000, 1334139660000000, 2134623456000000, 2195612697600000, 1505562992640000, 688257368064000, 202263389798400, 34673723965440, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (35+24*x)*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[(35 + 24*x)*P[n - 1, 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] = (35 + 24*x)*v[i - 1]); 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.