Real-rooted atlas

A038338 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(60 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3600 + 4320\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03600432012969216
121600038880023328046656884736
212960000311040002799360011197440167961684934656
3777600000233280000027993600001679616000503884800604661768153726976
44665600000016796160000025194240000020155392000090699264000217678233602176782336782757789696
527993600000001175731200000021163161600000211631616000001269789696000045712429056009142485811207836416409675144747810816
61679616000000008062156800000001693052928000000203166351360000015237476352000007313988648960002194196594688003761479876608028211099074567213895789838336
710077696000000000544195584000000001306069401600000001828497162240000001645647446016000009873884676096000039495538704384000101559956668416001523399350026240101559956668416692533995824480256

All 15 rows

Technical details
Node id
n-b885e5ca17d9686b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b1029f17d1b64336

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:                 3600,                  4320,                  1296;
  1:               216000,                388800,                233280,                  46656;
  2:             12960000,              31104000,              27993600,               11197440,                1679616;
  3:            777600000,            2332800000,            2799360000,             1679616000,              503884800,               60466176;
  4:          46656000000,          167961600000,          251942400000,           201553920000,            90699264000,            21767823360,            2176782336;
  5:        2799360000000,        11757312000000,        21163161600000,         21163161600000,         12697896960000,          4571242905600,          914248581120,           78364164096;
  6:      167961600000000,       806215680000000,      1693052928000000,       2031663513600000,       1523747635200000,        731398864896000,       219419659468800,        37614798766080,         2821109907456;
  7:    10077696000000000,     54419558400000000,    130606940160000000,     182849716224000000,     164564744601600000,      98738846760960000,     39495538704384000,     10155995666841600,      1523399350026240,      101559956668416;
  8:   604661760000000000,   3627970560000000000,   9795520512000000000,   15672832819200000000,   16456474460160000000,   11848661611315200000,   5924330805657600000,   2031199133368320000,    457019805007872000,    60935974001049600,    3656158440062976;
  9: 36279705600000000000, 239446056960000000000, 718338170880000000000, 1293008707584000000000, 1551610449100800000000, 1303352777244672000000, 782011666346803200000, 335147857005772800000, 100544357101731840000, 20108871420346368000, 2413064570441564160, 131621703842267136;
%F Formula
The row polynomials satisfy P(n,x) = (60+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 3600 + 4320*x + 1296*x^2.
%t Mathematica
P[0, x_] := 3600 + 4320*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(60 + 36*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] = 3600 + 4320*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 36*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.