Real-rooted atlas

A048854: Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].

A048854 ↗ nonnegative coefficients

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

Recurrence from A048854

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(8 + 8\,x\right)\,P_{n-1}'(x) + \left(16\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
31212125
4120180301331
5168033608405615937
63024075600252002520901133651
76652801995840831600110880594013213609673
81729728060540480302702405045040360360120121821113525595
951891840020756736001210809600242161920216216009609602184024014070168161
10176432256007939451520052929676800123502579201323241920735134402227680367203061163716695587
116704425728003352212864000251415964800067044257280083805321600558702144021162960046512005814038017296866339961
1228158588057600154872234316800129060195264000387180585792005531151225600430200650880195545750405372136008953560877804621356790528924523
13129529505064960077717703038976007124122778572800237470759285760038164943456640033924394183680179902090368059308381440123559128016151520127512552118983329135883665
1464764752532480000420970891461120000420970891461120000154355993535744000275635702742400002756357027424000167051941056000642507465600016062686640026246220002762760017940065011091556096587136051
15349729663675392000024481076457277440000265211661620505600001060846646482022400020838059127325440002315339903036160001578640842979200069390806284800020238985166400396842846400522161640045208800245700756167439846581526028457

b-files: A048854 ?

Technical details
Node id
n-be41df8499a2681b · revision 1 · JSON
Definition source
oeis_catalog:A048854
Fingerprint
f6-14fed6d19df1fb72

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 ?

  • A048854 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:           1;
   2:           2,           1;
   3:          12,          12,           1;
   4:         120,         180,          30,           1;
   5:        1680,        3360,         840,          56,          1;
   6:       30240,       75600,       25200,        2520,         90,        1;
   7:      665280,     1995840,      831600,      110880,       5940,      132,       1;
   8:    17297280,    60540480,    30270240,     5045040,     360360,    12012,     182,     1;
   9:   518918400,  2075673600,  1210809600,   242161920,   21621600,   960960,   21840,   240,   1;
  10: 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (8+8*x)*d/dx P(n-1,x) + 16*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[(2 + x)*P[n - 1, x] + (8 + 8*x)*D[P[n - 1, x], x] + 16*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] = (2 + x)*v[i - 1] + (8 + 8*x)*deriv(v[i - 1], x) + 16*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

A048854 triangle, offset 0, from P1
%e Example
Triangle begins:
  0:           1;
  1:           2,           1;
  2:          12,          12,           1;
  3:         120,         180,          30,           1;
  4:        1680,        3360,         840,          56,          1;
  5:       30240,       75600,       25200,        2520,         90,        1;
  6:      665280,     1995840,      831600,      110880,       5940,      132,       1;
  7:    17297280,    60540480,    30270240,     5045040,     360360,    12012,     182,     1;
  8:   518918400,  2075673600,  1210809600,   242161920,   21621600,   960960,   21840,   240,   1;
  9: 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (8+8*x)*d/dx P(n-1,x) + 16*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[(2 + x)*P[n - 1, x] + (8 + 8*x)*D[P[n - 1, x], x] + 16*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] = (2 + x)*v[i - 1] + (8 + 8*x)*deriv(v[i - 1], x) + 16*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 1

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.