Real-rooted atlas

A290598: Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,2].

A290598 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence from A290598

\(P_{n}(x) = \left(4 + x\right)\,P_{n-1}(x) + \left(12 + 6\,x\right)\,P_{n-1}'(x) + \left(9\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2415
32814143
4280210301521
5364036407805218113
65824072800208002080801154001
71106560165984059280079040456011413442915
824344320426025601825824030430402340808778154188491173
96086080001217216000608608000121721600117040005852001540020012568458401
10170410240003834230400021909888000511230720058988160036867600129360025200252183033591453
11528271744000132067936000084900816000022640217600030477216000228579120010025400026040003906031012957227344571
1217961239296000493934080640003528100576000010584301728000162835411200014248098480074989992002434740004869480579703741114998537345825
13664565853952000199369756185600015664766557440005221588852480009037365321600090373653216005549259408002162049120054051228085795608302844414846887070944913
14265826341580800008639356101376000074051623726080000271522620328960005221588852480000587428745904000412230698880001873775904000562132771201115342800143915201154405201220032252718119401
1511430532687974400004000686440791040000371492312359168000014859692494366720003143396489192960003929245611491200031020360090720001611447277440005640065471040134287273120216592376023164960156520602110701533144929846003

b-files: A290598 ?

Technical details
Node id
n-e18d516553291356 · revision 1 · JSON
Definition source
oeis_catalog:A290598
Fingerprint
f6-d99ad6edcacedc3c

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • 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): no

Matches ?

  • A290598 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:           1;
   2:           4,           1;
   3:          28,          14,           1;
   4:         280,         210,          30,          1;
   5:        3640,        3640,         780,         52,         1;
   6:       58240,       72800,       20800,       2080,        80,        1;
   7:     1106560,     1659840,      592800,      79040,      4560,      114,       1;
   8:    24344320,    42602560,    18258240,    3043040,    234080,     8778,     154,     1;
   9:   608608000,  1217216000,   608608000,  121721600,  11704000,   585200,   15400,   200,   1;
  10: 17041024000, 38342304000, 21909888000, 5112307200, 589881600, 36867600, 1293600, 25200, 252, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (12+6*x)*d/dx P(n-1,x) + 9*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + (12 + 6*x)*D[P[n - 1, x], x] + 9*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (4 + x)*v[i - 1] + (12 + 6*x)*deriv(v[i - 1], x) + 9*x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A290598 triangle, offset 0, from P1
%e Example
Triangle begins:
  0:           1;
  1:           4,           1;
  2:          28,          14,           1;
  3:         280,         210,          30,          1;
  4:        3640,        3640,         780,         52,         1;
  5:       58240,       72800,       20800,       2080,        80,        1;
  6:     1106560,     1659840,      592800,      79040,      4560,      114,       1;
  7:    24344320,    42602560,    18258240,    3043040,    234080,     8778,     154,     1;
  8:   608608000,  1217216000,   608608000,  121721600,  11704000,   585200,   15400,   200,   1;
  9: 17041024000, 38342304000, 21909888000, 5112307200, 589881600, 36867600, 1293600, 25200, 252, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (12+6*x)*d/dx P(n-1,x) + 9*x*d^2/dx^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[(4 + x)*P[n - 1, x] + (12 + 6*x)*D[P[n - 1, x], x] + 9*x*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (4 + x)*v[i - 1] + (12 + 6*x)*deriv(v[i - 1], x) + 9*x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 6

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.