Real-rooted atlas

A013612 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(26 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 676 + 1040\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
067610404002116
1175764056031200800097336
2456976140608016224008320001600004477456
311881376456976007030400054080000208000003200000205962976
43089157761425765120274185600028121600001622400000499200000640000009474296896
5803181017643248208640998035584001279532800009842560000045427200000116480000001280000000435817657216
62088270645761285089628160345985669120053228564480005118131200000314961920000012113920000002662400000002560000000020047612231936
754295036789763758887162368011565806653440020759140147200023952854016000018425272320000094488576000000311500800000005990400000000512000000000922190162669056

All 15 rows

Technical details
Node id
n-0176d4aaeb293b4e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-de8557183b81eda9

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,              1040,                400;
  1:            17576,             40560,              31200,               8000;
  2:           456976,           1406080,            1622400,             832000,             160000;
  3:         11881376,          45697600,           70304000,           54080000,           20800000,            3200000;
  4:        308915776,        1425765120,         2741856000,         2812160000,         1622400000,          499200000,           64000000;
  5:       8031810176,       43248208640,        99803558400,       127953280000,        98425600000,        45427200000,        11648000000,         1280000000;
  6:     208827064576,     1285089628160,      3459856691200,      5322856448000,      5118131200000,      3149619200000,      1211392000000,       266240000000,       25600000000;
  7:    5429503678976,    37588871623680,    115658066534400,    207591401472000,    239528540160000,    184252723200000,     94488576000000,     31150080000000,     5990400000000,      512000000000;
  8:  141167095653376,  1085900735795200,   3758887162368000,   7710537768960000,  10379570073600000,   9581141606400000,   6141757440000000,   2699673600000000,   778752000000000,   133120000000000,   10240000000000;
  9: 3670344486987776, 31056761043742720, 119449080937472000, 275651725240320000, 424079577292800000, 456701083238400000, 351308525568000000, 193026662400000000, 74241024000000000, 19036160000000000, 2928640000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = (26+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 1040*x + 400*x^2.
%t Mathematica
P[0, x_] := 676 + 1040*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 20*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 + 1040*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 20*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.