Real-rooted atlas

A038243 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(6 + 16\,x + 10\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161016
236120100256
32161080180010004096
41296864021600240001000065536
57776648002160003600003000001000001048576
6466564665601944000432000054000003600000100000016777216
72799363265920163296004536000075600000756000004200000010000000268435456
8167961622394880130636800435456000907200000120960000010080000004800000001000000004294967296
9100776961511654401007769600391910400097977600001632960000018144000000129600000005400000000100000000068719476736
106046617610077696007558272000335923200009797760000019595520000027216000000025920000000016200000000060000000000100000000001099511627776
113627970566651279360554273280002771366400009237888000002155507200000359251200000042768000000003564000000000198000000000066000000000010000000000017592186044416
122176782336435356467203990767616002217093120000831409920000022170931200000431101440000006158592000000064152000000000475200000000002376000000000072000000000001000000000000281474976710656
131306069401628298170368028298170368001729332633600072055526400000216166579200000480370176000000800616960000000100077120000000092664000000000061776000000000028080000000000078000000000000100000000000004503599627370496
14783641640961828497162240198087192576001320581283840006052664217600002017554739200000504388684800000096074035200000001401079680000000015567552000000000129729600000000007862400000000000327600000000000084000000000000010000000000000072057594037927936
Technical details
Node id
n-24fd2fda67fd61e1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b1029f17d1b64336

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:        6,        10;
  2:       36,       120,        100;
  3:      216,      1080,       1800,       1000;
  4:     1296,      8640,      21600,      24000,      10000;
  5:     7776,     64800,     216000,     360000,     300000,      100000;
  6:    46656,    466560,    1944000,    4320000,    5400000,     3600000,     1000000;
  7:   279936,   3265920,   16329600,   45360000,   75600000,    75600000,    42000000,    10000000;
  8:  1679616,  22394880,  130636800,  435456000,  907200000,  1209600000,  1008000000,   480000000,  100000000;
  9: 10077696, 151165440, 1007769600, 3919104000, 9797760000, 16329600000, 18144000000, 12960000000, 5400000000, 1000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (6+16*x+10*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[((6 + 16*x + 10*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] = ((6 + 16*x + 10*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.