Real-rooted atlas

A013609 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 10\,x + 7\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13710
294249100
3271894413431000
48175626464116240110000
5243283513230308703601516807100000
672910206595351852203241353025261176491000000
721873572125004797240522689453176523247062982354310000000
8656112247210001884667544136136702541218429647548197650325764801100000000
91968341334338578682100394873513818171532242266827932266827932155649627403536071000000000
1059049137781014467005900169203675690901029193452200120949026682793202334744405121060821028247524910000000000
11177147454677353045685371319795173282571056605639861320798263422013304390256821884551997503546593216832171977326743100000000000
125314411488034819096446614852791807797715695291114719287924789580415849579160823113969609523970042558016779029790671183762748138412872011000000000000
13159432348361131677055834579258880233790101345141918425649441523990908103022264545218028896295412337079149405218127387277813880833735865398102008399688901040710000000000000
1447829691562436542369695419221171572441419184256496622859863622318000952267618133587271212620227406787196314648550022290337566416919433167230204113360142176194069338437094678223072849100000000000000
Technical details
Node id
n-ca0022e1bdb371d2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5184fe74b2189306

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:     3,      7;
  2:     9,     42,      49;
  3:    27,    189,     441,      343;
  4:    81,    756,    2646,     4116,     2401;
  5:   243,   2835,   13230,    30870,    36015,     16807;
  6:   729,  10206,   59535,   185220,   324135,    302526,    117649;
  7:  2187,  35721,  250047,   972405,  2268945,   3176523,   2470629,    823543;
  8:  6561, 122472, 1000188,  4667544, 13613670,  25412184,  29647548,  19765032,   5764801;
  9: 19683, 413343, 3857868, 21003948, 73513818, 171532242, 266827932, 266827932, 155649627, 40353607;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3+10*x+7*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[((3 + 10*x + 7*x^2)*P[n - 1, x])/(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] = ((3 + 10*x + 7*x^2)*v[i - 1])/(1 + 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

Candidate equalities ?

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.