Real-rooted atlas

A038319 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01296432036009216
146656233280388800216000884736
216796161119744027993600311040001296000084934656
3604661765038848001679616000279936000023328000007776000008153726976
42176782336217678233609069926400020155392000025194240000016796160000046656000000782757789696
578364164096914248581120457124290560012697896960000211631616000002116316160000011757312000000279936000000075144747810816
62821109907456376147987660802194196594688007313988648960001523747635200000203166351360000016930529280000008062156800000001679616000000007213895789838336
710155995666841615233993500262401015599566684160039495538704384000987388467609600001645647446016000001828497162240000001306069401600000005441955840000000010077696000000000692533995824480256

All 15 rows

Technical details
Node id
n-263d5a8e971d7391 · revision 0 · JSON
Definition source
derived:basis_map_A013610
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:               1296,                4320,                 3600;
  1:              46656,              233280,               388800,                216000;
  2:            1679616,            11197440,             27993600,              31104000,              12960000;
  3:           60466176,           503884800,           1679616000,            2799360000,            2332800000,             777600000;
  4:         2176782336,         21767823360,          90699264000,          201553920000,          251942400000,          167961600000,            46656000000;
  5:        78364164096,        914248581120,        4571242905600,        12697896960000,        21163161600000,        21163161600000,         11757312000000,          2799360000000;
  6:      2821109907456,      37614798766080,      219419659468800,       731398864896000,      1523747635200000,      2031663513600000,       1693052928000000,        806215680000000,        167961600000000;
  7:    101559956668416,    1523399350026240,    10155995666841600,     39495538704384000,     98738846760960000,    164564744601600000,     182849716224000000,     130606940160000000,      54419558400000000,     10077696000000000;
  8:   3656158440062976,   60935974001049600,   457019805007872000,   2031199133368320000,   5924330805657600000,  11848661611315200000,   16456474460160000000,   15672832819200000000,    9795520512000000000,   3627970560000000000,    604661760000000000;
  9: 131621703842267136, 2413064570441564160, 20108871420346368000, 100544357101731840000, 335147857005772800000, 782011666346803200000, 1303352777244672000000, 1551610449100800000000, 1293008707584000000000, 718338170880000000000, 239446056960000000000, 36279705600000000000;
%F Formula
The row polynomials satisfy P(n,x) = (36+60*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 4320*x + 3600*x^2.
%t Mathematica
P[0, x_] := 1296 + 4320*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 60*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] = 1296 + 4320*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 60*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.