Real-rooted atlas

A013620 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 19\,x + 14\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151419
225140196361
31251050294027446859
46257000294005488038416130321
53125437502450006860009604005378242476099
615625262500183750068600001440600016134720752953647045881
77812515312501286250060025000168070000282357600263533760105413504893871739
83906258750000857500004802000001680700000376476800052706752004216540160147578905616983563041
91953125492187505512500003601500000151263000004235364000079060128000948721536006641050752020661046784322687697779
1097656252734375003445312500257250000001260525000004235364000009882516000001581202560000166026268800010330523392002892546549766131066257801
1148828125150390625021054687500176859375000990412500000388241700000010870767600000217415352000003043814928000028408939328000159090060236804049565169664116490258898219
1224414062582031250001263281250001179062500000742809375000033277860000000108707676000000260898422400000456572239200000568178786560000477270180710400242973910179840566939123752962213314919066161
131220703125444335937507464843750007663906250000536473437500002703826125000001009428420000000282639957600000059354391096000009232905281600000103408539153920007896652080844800368510430439424079371477325414442052983462257059
1461035156252392578125004354492187500487703125000003755314062500002102975875000000883249867500000028263995760000000692467896120000001292606739424000001809649435193600001842552152197120001289786506537984005556003412779008011112006825558016799006685782884121
Technical details
Node id
n-273e5590e99fa0f3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ec7576139aa067f7

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:       5,       14;
  2:      25,      140,       196;
  3:     125,     1050,      2940,       2744;
  4:     625,     7000,     29400,      54880,       38416;
  5:    3125,    43750,    245000,     686000,      960400,      537824;
  6:   15625,   262500,   1837500,    6860000,    14406000,    16134720,     7529536;
  7:   78125,  1531250,  12862500,   60025000,   168070000,   282357600,   263533760,   105413504;
  8:  390625,  8750000,  85750000,  480200000,  1680700000,  3764768000,  5270675200,  4216540160,  1475789056;
  9: 1953125, 49218750, 551250000, 3601500000, 15126300000, 42353640000, 79060128000, 94872153600, 66410507520, 20661046784;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (5+19*x+14*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[((5 + 19*x + 14*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] = ((5 + 19*x + 14*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

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.