Real-rooted atlas

A038301 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(31 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(840 + 682\,x + 121\,x^{2} + 220\,n + 121\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 840 + 682\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
08406821211643
1212002772011253133161504
2512000932800609840165044146412234325
312000000281600002565200011180400226935516105179422806
42752000007920000009292800005643440001844766002995548617715612777027647
562080000002119040000030492000000238515200001086362200028409396403844287371948717195850397548
61382400000005463040000009323776000008944320000005247334400001911997472004166711472048328184082143588813274001079209
7304640000000013685760000000270420480000003076846080000022137192000000103897221120003154795828800589292051040598061277992357947691110875834867330

All 15 rows

Technical details
Node id
n-eb79832434990a0c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bd6aa0c340783573

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:              840,              682,               121;
  1:            21200,            27720,             11253,              1331;
  2:           512000,           932800,            609840,            165044,             14641;
  3:         12000000,         28160000,          25652000,          11180400,           2269355,            161051;
  4:        275200000,        792000000,         929280000,         564344000,         184476600,          29955486,          1771561;
  5:       6208000000,      21190400000,       30492000000,       23851520000,       10863622000,        2840939640,        384428737,         19487171;
  6:     138240000000,     546304000000,      932377600000,      894432000000,      524733440000,      191199747200,      41667114720,       4832818408,       214358881;
  7:    3046400000000,   13685760000000,    27042048000000,    30768460800000,    22137192000000,    10389722112000,    3154795828800,     589292051040,     59806127799,      2357947691;
  8:   66560000000000,  335104000000000,   752716800000000,   991541760000000,   846132672000000,   487018224000000,  190478238720000,   49575363024000,   8102765701800,    730963784210,   25937424601;
  9: 1443840000000000, 8053760000000000, 20273792000000000, 30359577600000000, 29994138240000000, 20476410662400000, 9821534184000000, 3292552412160000, 749827365738000, 108937183324200, 8844661788941, 285311670611;
%F Formula
The row polynomials satisfy (11*n+31+11*x)*P(n,x) = (220*n+840+(121*n+682)*x+121*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 840 + 682*x + 121*x^2.
%t Mathematica
P[0, x_] := 840 + 682*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((220*n + 840 + (121*n + 682)*x + 121*x^2)*P[n - 1, x])/(11*n + 31 + 11*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 840 + 682*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((220*n + 840 + (121*n + 682)*x + 121*x^2)*v[i - 1])/(11*n + 31 + 11*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. [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.