Real-rooted atlas

A038277 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(40 + 44\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1404484
21600352019367056
36400021120023232085184592704
42560000112640001858560013629440374809649787136
5102400000563200000123904000013629440007496192001649162244182119424
64096000000270336000007434240000010903552000089954304000395798937607256313856351298031616
716384000000012615680000004163174400000763248640000083957350400005541185126400203176787968031927780966429509034655744

All 15 rows

Technical details
Node id
n-4cca32ac9f07ae5b · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-c4264000c5d1dd22

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:              40,               44;
  2:            1600,             3520,              1936;
  3:           64000,           211200,            232320,             85184;
  4:         2560000,         11264000,          18585600,          13629440,           3748096;
  5:       102400000,        563200000,        1239040000,        1362944000,         749619200,         164916224;
  6:      4096000000,      27033600000,       74342400000,      109035520000,       89954304000,       39579893760,        7256313856;
  7:    163840000000,    1261568000000,     4163174400000,     7632486400000,     8395735040000,     5541185126400,     2031767879680,      319277809664;
  8:   6553600000000,   57671680000000,   222035968000000,   488479129600000,   671658803200000,   591059746816000,   325082860748800,   102168899092480,   14048223625216;
  9: 262144000000000, 2595225600000000, 11418992640000000, 29308747776000000, 48359433830400000, 53195377213440000, 39009943289856000, 18390401836646400, 5057360505077760, 618121839509504;
%F Formula
The row polynomials satisfy P(n,x) = (40+44*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[(40 + 44*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] = (40 + 44*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.