Real-rooted atlas

A166962: Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.

A166962 ↗ nonnegative coefficients

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

Recurrence from A166962

\(P_{n}(x) = \left(1 - 2\,x + 3\,n x\right)\,P_{n-1}(x) + \left(7\,x - 3\,x^{2}\right)\,P_{n-1}'(x) + \left(3\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
2112114
31103691174
41834217031613322
51668553910278301329189756
615349612190151652300281195541213211420
714279872645525181939195348006752487917217811146132808
81342391856369202437008518163327253410586846782203122928730018202467544
9127391369119128587001587442210962754851007069959158540285700026561575508443494251554489060200
10121913098025085481692566118517352402092267174671413968553809888243450096347411342897510411814765331397980144303264052840
11117530478715274086822865270745439832725150543440171169017385119289368145472854891002505161201214346614001219709658661770131559226114120273900929040
1211402438300211081016776416610977492024349220104621872107077975200016702015591324105820696700132716178402819628540169859836013321831642651534633062539351918223694521440390341680552304
13111219506405323274903520695544425347961746579467107791463641846535218171501961737164791847452010546066035651504828627648284205072908217054054311741721111429077918559107246733844690219984289478289153519701097646685024
1418975605124644888144861083059517780526518956682676475725023818134421101229346457981464848496430025399289306107393231312924426166744329752211999609276247961955129934722592879050162897103001971423852038246822208916317135445590335791371617327768200849533588768

b-files: A166962 ?

Technical details
Node id
n-25ac1cc25c118491 · revision 0 · JSON
Definition source
oeis_catalog:A166962
Fingerprint
f6-53c493e1859e92ba

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 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): no

Matches ?

  • A166962 triangle 41 terms, 8 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,        1;
  2: 1,       12,           1;
  3: 1,      103,          69,            1;
  4: 1,      834,        2170,          316,            1;
  5: 1,     6685,       53910,        27830,         1329,           1;
  6: 1,    53496,     1219015,      1652300,       281195,        5412,          1;
  7: 1,   427987,    26455251,     81939195,     34800675,     2487917,      21781,         1;
  8: 1,  3423918,   563692024,   3700851816,   3327253410,   586846782,   20312292,     87300,      1;
  9: 1, 27391369, 11912858700, 158744221096, 275485100706, 99591585402, 8570002656, 157550844, 349425, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n-2)*x)*P(n-1,x) + (7*x-3*x^2)*d/dx P(n-1,x) + 3*x^2*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[(1 + (3*n - 2)*x)*P[n - 1, x] + (7*x - 3*x^2)*D[P[n - 1, x], x] + 3*x^2*D[P[n - 1, x], {x, 2}]];
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] = (1 + (3*n - 2)*x)*v[i - 1] + (7*x - 3*x^2)*deriv(v[i - 1], x) + 3*x^2*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 - 1, v[i])))};
T(9)

polytool c48629374597

A166962 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       12,           1;
   4: 1,      103,          69,            1;
   5: 1,      834,        2170,          316,            1;
   6: 1,     6685,       53910,        27830,         1329,           1;
   7: 1,    53496,     1219015,      1652300,       281195,        5412,          1;
   8: 1,   427987,    26455251,     81939195,     34800675,     2487917,      21781,         1;
   9: 1,  3423918,   563692024,   3700851816,   3327253410,   586846782,   20312292,     87300,      1;
  10: 1, 27391369, 11912858700, 158744221096, 275485100706, 99591585402, 8570002656, 157550844, 349425, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n-5)*x)*P(n-1,x) + (7*x-3*x^2)*d/dx P(n-1,x) + 3*x^2*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[(1 + (3*n - 5)*x)*P[n - 1, x] + (7*x - 3*x^2)*D[P[n - 1, x], x] + 3*x^2*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n], {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] = (1 + (3*n - 5)*x)*v[i - 1] + (7*x - 3*x^2)*deriv(v[i - 1], x) + 3*x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 7

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.