Real-rooted atlas

A137320 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 4\,x + 3\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26222048
324124212120480
4120788192820848405760
57205688178722792821712672080640
6504046296176296356344403336242448604801290240
7403204207681873328461371267901285972944290832060480023224320
83628804230432214884006213001611186198412844790491877264374347206652800464486400
936288004665888026564918487916096018641800322626822848246014748814768743685157446407983360010218700800
1039916800560422080352870646413124209952319350728325312939174461210319392482275353922487255782475828499201037836800245248819200
11479001600728390016050190386688206892409920566959813312108464372057614783353171201435674896192973656189376439210008576118613940480145297152006376469299200
126227020800101875726080761733529344344245712928010473863721856226047655671683548801493120040838803407296341926539047682031457295212881301313548801968095404800217945728000178541140377600
138717829120015258924979201229428102809660382136279424201713406174464484048537490048858508458111488113955148660134411321180091834888314864838060804388550062023681576354373452803454076666880034871316480005356234211328000
14130767436800024370418419200210354387886080111473482166899240521974093671681068985596731660821106452009003136316879160869154563635414540997516831718303413210496207180953177388809825066665619456319791393490176063950000808960059281238016000171399494762496000
Technical details
Node id
n-c8711ac6278fb659 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-58bd42dcb256e0bd

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:       2,        4;
  2:       6,       22,        20;
  3:      24,      124,       212,       120;
  4:     120,      788,      1928,      2084,        840;
  5:     720,     5688,     17872,     27928,      21712,       6720;
  6:    5040,    46296,    176296,    356344,     403336,     242448,      60480;
  7:   40320,   420768,   1873328,   4613712,    6790128,    5972944,    2908320,     604800;
  8:  362880,  4230432,  21488400,  62130016,  111861984,  128447904,   91877264,   37434720,   6652800;
  9: 3628800, 46658880, 265649184, 879160960, 1864180032, 2626822848, 2460147488, 1476874368, 515744640, 79833600;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+1+(2*n+4)*x+(n+3)*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[((n + 1 + (2*n + 4)*x + (n + 3)*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] = ((n + 1 + (2*n + 4)*x + (n + 3)*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

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.