Real-rooted atlas

A038321 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(25 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 625 + 350\,x + 49\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0625350491024
11562513125367534332768
23906254375001837503430024011048576
3976562513671875765625021437503001251680733554432
42441406254101562502871093751071875002250937525210501176491073741824
561035156251196289062510048828125468945312513130468752205918752058857582354334359738368
61525878906253417968750003349609375001875781250006565234375014706125000205885750016470860057648011099511627776
73814697265625961303710937510766601562500703417968750029543554687508272195312501544143125001852971750012970802254035360735184372088832

All 15 rows

Technical details
Node id
n-1b09cdcaf124bad8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b7480fed0cfd4971

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:              625,              350,                49;
  1:            15625,            13125,              3675,              343;
  2:           390625,           437500,            183750,            34300,             2401;
  3:          9765625,         13671875,           7656250,          2143750,           300125,            16807;
  4:        244140625,        410156250,         287109375,        107187500,         22509375,          2521050,          117649;
  5:       6103515625,      11962890625,       10048828125,       4689453125,       1313046875,        220591875,        20588575,          823543;
  6:     152587890625,     341796875000,      334960937500,     187578125000,      65652343750,      14706125000,      2058857500,       164708600,        5764801;
  7:    3814697265625,    9613037109375,    10766601562500,    7034179687500,    2954355468750,     827219531250,    154414312500,     18529717500,     1297080225,      40353607;
  8:   95367431640625,  267028808593750,   336456298828125,  251220703125000,  123098144531250,   41360976562500,   9650894531250,   1544143125000,   162135028125,   10088401750,   282475249;
  9: 2384185791015625, 7343292236328125, 10280609130859375, 8635711669921875, 4835998535156250, 1895711425781250, 530799199218750, 106159839843750, 14862377578125, 1387155240625, 77680693475, 1977326743;
%F Formula
The row polynomials satisfy P(n,x) = (25+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 625 + 350*x + 49*x^2.
%t Mathematica
P[0, x_] := 625 + 350*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(25 + 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] = 625 + 350*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (25 + 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

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.