Real-rooted atlas

A038333 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(26 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 676 + 364\,x + 49\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0676364491089
11757614196382234335937
24569764921281987443567224011185921
31188137615994160861224023186803121301680739135393
43089157764990177923358773601205713602434614026218921176491291467969
580318101761513687302412225935904548599688014769991602385921722141211882354342618442977
62088270645764497813698564238324446722282174702087680395632016542390592222686027217129694457648011406408618241
754295036789761315610506828814168113150464890048133811235944251557769677298496321736951012162004174244813489634344035360746411484401953

All 15 rows

Technical details
Node id
n-9034373aeaf0a674 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8768456193250bc4

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:              676,               364,                49;
  1:            17576,             14196,              3822,               343;
  2:           456976,            492128,            198744,             35672,             2401;
  3:         11881376,          15994160,           8612240,           2318680,           312130,            16807;
  4:        308915776,         499017792,         335877360,         120571360,         24346140,          2621892,          117649;
  5:       8031810176,       15136873024,       12225935904,        5485996880,       1476999160,        238592172,        21412118,          823543;
  6:     208827064576,      449781369856,      423832444672,      228217470208,      76803956320,      16542390592,      2226860272,       171296944,        5764801;
  7:    5429503678976,    13156105068288,    14168113150464,     8900481338112,    3594425155776,     967729849632,    173695101216,     20041742448,     1348963434,      40353607;
  8:  141167095653376,   380065257528320,   460463677390080,   330589306844160,  155758423416960,   50321952180864,  11290181579040,   1736951012160,   175365246420,   10491937820,   282475249;
  9: 3670344486987776, 10869866365309952, 14632512414840320, 11818567719678720, 6363844156750080, 2398679720621184, 645798386321088, 124191997369440, 16718153492040, 1500347108260, 80787921214, 1977326743;
%F Formula
The row polynomials satisfy P(n,x) = (26+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 364*x + 49*x^2.
%t Mathematica
P[0, x_] := 676 + 364*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 7*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] = 676 + 364*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 7*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

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.