Real-rooted atlas

A139548 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(50 + 120\,x + 72\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15012072242
22500120002160017280518458564
3125000900000270000043200003888000186624037324814172488
4625000060000000252000000604800000907200000870912000522547200179159040268738563429742096
53125000003750000000202500000006480000000013608000000019595520000019595520000013436928000060466176000161243136001934917632829997587232
6156250000002250000000001485000000000594000000000016038000000000307929600000004311014400000044341862400000332563968000001773674496000063852281856001393140695040139314069504200859416110144
7781250000000131250000000001023750000000004914000000000001621620000000000389188800000000070053984000000009607403520000000100877736960000008070218956800000484213137408000021129300541440006338790162432001170238183833601003061300428848607978698654848

All 15 rows

Technical details
Node id
n-3b4b7a0ddd26d13e · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-0d4cb94e85dd3af7

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:                1;
  1:               50,               120,                 72;
  2:             2500,             12000,              21600,               17280,                 5184;
  3:           125000,            900000,            2700000,             4320000,              3888000,              1866240,                373248;
  4:          6250000,          60000000,          252000000,           604800000,            907200000,            870912000,             522547200,             179159040,              26873856;
  5:        312500000,        3750000000,        20250000000,         64800000000,         136080000000,         195955200000,          195955200000,          134369280000,           60466176000,           16124313600,            1934917632;
  6:      15625000000,      225000000000,      1485000000000,       5940000000000,       16038000000000,       30792960000000,        43110144000000,        44341862400000,        33256396800000,        17736744960000,         6385228185600,         1393140695040,          139314069504;
  7:     781250000000,    13125000000000,    102375000000000,     491400000000000,     1621620000000000,     3891888000000000,      7005398400000000,      9607403520000000,     10087773696000000,      8070218956800000,      4842131374080000,      2112930054144000,       633879016243200,       117023818383360,       10030613004288;
  8:   39062500000000,   750000000000000,   6750000000000000,   37800000000000000,   147420000000000000,   424569600000000000,    934053120000000000,   1601233920000000000,   2161665792000000000,   2305776844800000000,   1936852549632000000,   1267758032486400000,    633879016243200000,    234047636766720000,    60183678025728000,     9629388484116480,     722204136308736;
  9: 1953125000000000, 42187500000000000, 430312500000000000, 2754000000000000000, 12393000000000000000, 41640480000000000000, 108265248000000000000, 222717081600000000000, 367483184640000000000, 489977579520000000000, 529175785881600000000, 461826140405760000000, 323278298284032000000, 179046442126540800000, 76734189482803200000, 24554940634497024000, 5524861642761830400, 779980467213434880, 51998697814228992;
%F Formula
The row polynomials satisfy P(n,x) = (50+120*x+72*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(50 + 120*x + 72*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (50 + 120*x + 72*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.