Real-rooted atlas

A038331 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 39\,x + 22\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 748\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02897484841521
1491319074246841064859319
2835214323448392567240642342562313441
314198579187310237789203077272019911760515363290224199
424137569187421124606362460104627248010154997605256704641133799043518743761
5410338673371718562614431426548311266062804028149048031277392608134922085762494357888137231006679
669757574417221960644832711233508884664369081613695706763201417908464896917470183168339232672768548758735365352009260481
711858787649713811999733187149741038352215894141158084190886269539254234998782272467909793415682595129946675283960086510081207269217792208728361158759

All 15 rows

Technical details
Node id
n-ede5df9c71a22b22 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6f86be61a76d4bf4

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:            289,             748,              484;
  1:           4913,           19074,            24684,             10648;
  2:          83521,          432344,           839256,            724064,            234256;
  3:        1419857,         9187310,         23778920,          30772720,          19911760,           5153632;
  4:       24137569,       187421124,        606362460,        1046272480,        1015499760,         525670464,         113379904;
  5:      410338673,      3717185626,      14431426548,       31126606280,       40281490480,       31277392608,       13492208576,        2494357888;
  6:     6975757441,     72219606448,     327112335088,      846643690816,     1369570676320,     1417908464896,      917470183168,      339232672768,       54875873536;
  7:   118587876497,   1381199973318,    7149741038352,    21589414115808,    41908862695392,    54234998782272,    46790979341568,    25951299466752,     8396008651008,     1207269217792;
  8:  2015993900449,  26089332829340,  151931997064980,   524314342812480,  1187417776369440,  1843989958597248,  1988616622016640,  1470573636449280,   713660735335680,   205235767024640,   26559922791424;
  9: 34271896307633, 487870523908658, 3156809272350140, 12255847763241720, 31721017740155040, 57471020376280896, 74374261663422336, 68749317504003840, 44484852502590720, 19189544216803840, 4966705561996288, 584318301411328;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (17+39*x+22*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 748*x + 484*x^2.
%t Mathematica
P[0, x_] := 289 + 748*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 39*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] = 289 + 748*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 39*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

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.