Real-rooted atlas

A038277 → x ↦ x/(x+1) → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(7 + 32\,x + 25\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
172532
2493506251024
33433675131251562532768
42401343001837504375003906251048576
5168073001252143750765625013671875976562533554432
61176492521050225093751071875002871093754101562502441406251073741824
782354320588575220591875131304687546894531251004882812511962890625610351562534359738368
85764801164708600205885750014706125000656523437501875781250003349609375003417968750001525878906251099511627776
94035360712970802251852971750015441431250082721953125029543554687507034179687500107666015625009613037109375381469726562535184372088832
10282475249100884017501621350281251544143125000965089453125041360976562500123098144531250251220703125000336456298828125267028808593750953674316406251125899906842624
11197732674377680693475138715524062514862377578125106159839843750530799199218750189571142578125048359985351562508635711669921875102806091308593757343292236328125238418579101562536028797018963968
121384128720159319802290011652104021250138715524062500111467831835937563695903906250002653995996093750081244775390625000181349945068359375287857055664062500308418273925781250200271606445312500596046447753906251152921504606846976
139688901040744984183403259639467872125012623112689687501127063633007812572454090693359375345019479492187500123221242675781250033005690002441406256548748016357421875935535430908203125091123580932617187505424022674560546875149011611938476562536893488147419103232
14678223072849339111536424507872232095568751124604585081250011045223603476562578894454310546875042264886237792968751725097397460937500053909293670654296875128355461120605468750229206180572509765625297670364379882812500265777111053466796875146031379699707031250372529029846191406251180591620717411303424
Technical details
Node id
n-66be1d983d3de2fa · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b7480fed0cfd4971

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:        7,         25;
  2:       49,        350,         625;
  3:      343,       3675,       13125,        15625;
  4:     2401,      34300,      183750,       437500,       390625;
  5:    16807,     300125,     2143750,      7656250,     13671875,       9765625;
  6:   117649,    2521050,    22509375,    107187500,    287109375,     410156250,     244140625;
  7:   823543,   20588575,   220591875,   1313046875,   4689453125,   10048828125,   11962890625,     6103515625;
  8:  5764801,  164708600,  2058857500,  14706125000,  65652343750,  187578125000,  334960937500,   341796875000,  152587890625;
  9: 40353607, 1297080225, 18529717500, 154414312500, 827219531250, 2954355468750, 7034179687500, 10766601562500, 9613037109375, 3814697265625;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (7+32*x+25*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[((7 + 32*x + 25*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] = ((7 + 32*x + 25*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

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.