Real-rooted atlas

A038314 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(10 + 32\,x + 22\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 100 + 440\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01004404841024
110006600145201064832768
210000880002904004259202342561048576
3100000110000048400001064800011712800515363233554432
4100000013200000726000002129600003513840003092179201133799041073741824
510000000154000000101640000037268000008198960000108226272007936593280249435788834359738368
610000000017600000001355200000059628800000163979200000288603392000317463731200199548631040548758735361099511627776
710000000001980000000017424000000089443200000029516256000006493576320000952391193600089796883968004938828618240120726921779235184372088832

All 15 rows

Technical details
Node id
n-a85b59eef3df50f5 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2235b7dd30e0ad6a

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          100,           440,            484;
  1:         1000,          6600,          14520,           10648;
  2:        10000,         88000,         290400,          425920,          234256;
  3:       100000,       1100000,        4840000,        10648000,        11712800,          5153632;
  4:      1000000,      13200000,       72600000,       212960000,       351384000,        309217920,        113379904;
  5:     10000000,     154000000,     1016400000,      3726800000,      8198960000,      10822627200,       7936593280,       2494357888;
  6:    100000000,    1760000000,    13552000000,     59628800000,    163979200000,     288603392000,     317463731200,     199548631040,      54875873536;
  7:   1000000000,   19800000000,   174240000000,    894432000000,   2951625600000,    6493576320000,    9523911936000,    8979688396800,    4938828618240,    1207269217792;
  8:  10000000000,  220000000000,  2178000000000,  12777600000000,  49193760000000,  129871526400000,  238097798400000,  299322946560000,  246941430912000,  120726921779200,   26559922791424;
  9: 100000000000, 2420000000000, 26620000000000, 175692000000000, 773044800000000, 2380977984000000, 5238151564800000, 8231381030400000, 9054519133440000, 6639980697856000, 2921591507056640, 584318301411328;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (10+32*x+22*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 440*x + 484*x^2.
%t Mathematica
P[0, x_] := 100 + 440*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((10 + 32*x + 22*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] = 100 + 440*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((10 + 32*x + 22*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.