Real-rooted atlas

A120910 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(7 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
033
1211233
214716848363
31029176410081923993
472031646414112537676843923
55042114406016464094080268803072483153
6352947121010417287201317120564480129024122885314683
7247062998825161694145616134720921984031610886021124915258461513
8172944037906012815812025618070886412907776059006976168591362752512196608643076643
912106082162259850814230823041897443072162637977692935987235404185686704128123863047864327073843073
108474257474842432840124519701601897443072018974430720130110382086195732480202309632043352064055050240314572877812273803
11593198022937286732868106533522480182628895680208718737920166974990336954142801923894460416011127029760211943424024222105612582912855935011833
1241523861603284735050992894881588832170453635968021915467481602003699884032133579992268865426934988823366762496059344158720101732843521056964608503316489415285130163
132906670312212159240803356740311132579215511280873088221589726758402279208618086417365398994944992308513996842527507742721350079610880308589625344480918896644580179968201326592103568136431793
1420346692185471627735374837660458742493968138191411414784217157932223232248180493969408212726137688064138923191959552694615959797762646156037324875604458209281571001729024224428818432197300060168053063681139249500749723
Technical details
Node id
n-eae311e6cee1a668 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d36c4f1a2e9bd6b7

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:         3;
  1:        21,        12;
  2:       147,       168,         48;
  3:      1029,      1764,       1008,        192;
  4:      7203,     16464,      14112,       5376,        768;
  5:     50421,    144060,     164640,      94080,      26880,      3072;
  6:    352947,   1210104,    1728720,    1317120,     564480,    129024,     12288;
  7:   2470629,   9882516,   16941456,   16134720,    9219840,   3161088,    602112,    49152;
  8:  17294403,  79060128,  158120256,  180708864,  129077760,  59006976,  16859136,  2752512,   196608;
  9: 121060821, 622598508, 1423082304, 1897443072, 1626379776, 929359872, 354041856, 86704128, 12386304, 786432;
%F Formula
The row polynomials satisfy P(n,x) = (7+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(7 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = (7 + 4*x)*v[i - 1]); 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.