Real-rooted atlas

A144445 → D+1

nonnegative coefficients

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

Recurrence

\(\left(56\right)\,P_{n}(x) = \left(5432 - 1400\,x - 4802\,x^{2} - 29988\,x^{3} - 6958\,n - 3479\,n x + 31556\,n x^{3} + 2058\,n^{2} + 1029\,n^{2} x - 7203\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(2716\,x - 2058\,x^{2} - 29988\,x^{3} - 3479\,n x + 31556\,n x^{3} + 1029\,n^{2} x - 7203\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-20564 + 24230\,x + 18648\,x^{2} + 80192\,x^{3} + 31773\,n - 2534\,n x - 3038\,n x^{2} - 120148\,n x^{3} - 14749\,n^{2} - 4361\,n^{2} x - 1029\,n^{2} x^{2} + 52136\,n^{2} x^{3} + 2058\,n^{3} + 1029\,n^{3} x - 7203\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(5820 - 20318\,x + 5054\,x^{2} + 117726\,x^{3} + 29988\,x^{4} - 7455\,n + 23093\,n x + 5929\,n x^{2} - 151704\,n x^{3} - 31556\,n x^{4} + 2205\,n^{2} - 9506\,n^{2} x - 2058\,n^{2} x^{2} + 59339\,n^{2} x^{3} + 7203\,n^{2} x^{4} + 1029\,n^{3} x - 7203\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(2716\,x - 4774\,x^{2} - 27930\,x^{3} + 29988\,x^{4} - 3479\,n x + 3479\,n x^{2} + 31556\,n x^{3} - 31556\,n x^{4} + 1029\,n^{2} x - 1029\,n^{2} x^{2} - 7203\,n^{2} x^{3} + 7203\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(2748 - 27930\,x - 13608\,x^{2} + 10192\,x^{3} - 1855\,n + 24794\,n x + 9800\,n x^{2} - 5488\,n x^{3} - 147\,n^{2} - 4116\,n^{2} x - 1715\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-388 + 8918\,x - 8456\,x^{2} - 1372\,x^{3} + 5488\,x^{4} + 497\,n - 2058\,n x - 2450\,n x^{2} - 147\,n^{2} + 1029\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(2744\,x^{2} - 5488\,x^{3} + 2744\,x^{4}\right)\,P_{n-2}''(x) + \left(56 - 2828\,x + 784\,x^{2} + 294\,n + 784\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 24 + 25\,x + x^{2}\)
\(P_{3}(x) = 207 + 618\,x + 209\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
22425150
320761820911035
4167814359113721681129091
5134532803545338041872591345711028328
6107660491488320152305155306792941907107665143755100
78613238090945864241202910039799354009163274519200086132912175132402
868906341283193591182637618175262552689841063598093970411232068638591068906411123640359905
95512512919900791306480848306744236438612185333111164459521493703688302226434989508103638341135512513717906864428045
1044100109630450672721912014219427881951332129960532218708397892701765839366291355047322301985651703250927691560018729174410011051561707147557302
1135280088394621733792650289483206396395353512781666916312946782855871845169217049077952338436341867526069162564838822055411650662719531723443829802863528008849143878568842465201
122822407079069808462098295679926084148276512393342619796481268130972604954344713792216425956337331119420032081624882373535285326335215507127711566924882603691430960534352005837924322822407080113737640901174270195
1322579256640510513817101458101568218203637673824160685783508835913331588867623864028959929015830863153793812381841374394171048066888085723394754434156706688317664026231548134233693386435578801050150126265282874695865512257925664171344751315433466675910
14180634053133215807924977502147357013832708407366113519561420884425105515199963416826423035366493721194250245908106118770487815726261822309811083837415136982575370842677993100166609763231936434618831205246563579928371087337128224846774368764779265417453927111806340531345134229231618209288411832
Technical details
Node id
n-581672be31678c72 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0495af6a3fa95a19

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 ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        2,           1;
  2:       24,          25,            1;
  3:      207,         618,          209,             1;
  4:     1678,       14359,        11372,          1681,             1;
  5:    13453,      280354,       533804,        187259,         13457,             1;
  6:   107660,     4914883,     20152305,      15530679,       2941907,        107665,            1;
  7:   861323,    80909458,    642412029,    1003979935,     400916327,      45192000,       861329,           1;
  8:  6890634,  1283193591,  18263761817,   52625526898,   41063598093,    9704112320,    686385910,     6890641,        1;
  9: 55125129, 19900791306, 480848306744, 2364386121853, 3311116445952, 1493703688302, 226434989508, 10363834113, 55125137, 1;
%F Formula
The row polynomials satisfy 56*P(n,x) = (2058*n^2-6958*n+5432+(1029*n^2-3479*n-1400)*x-4802*x^2-(7203*n^2-31556*n+29988)*x^3)*P(n-4,x) + ((1029*n^2-3479*n+2716)*x-2058*x^2-(7203*n^2-31556*n+29988)*x^3)*d/dx P(n-4,x) + (2058*n^3-14749*n^2+31773*n-20564+(1029*n^3-4361*n^2-2534*n+24230)*x-(1029*n^2+3038*n-18648)*x^2-(7203*n^3-52136*n^2+120148*n-80192)*x^3)*P(n-3,x) + (2205*n^2-7455*n+5820+(1029*n^3-9506*n^2+23093*n-20318)*x-(2058*n^2-5929*n-5054)*x^2-(7203*n^3-59339*n^2+151704*n-117726)*x^3+(7203*n^2-31556*n+29988)*x^4)*d/dx P(n-3,x) + ((1029*n^2-3479*n+2716)*x-(1029*n^2-3479*n+4774)*x^2-(7203*n^2-31556*n+27930)*x^3+(7203*n^2-31556*n+29988)*x^4)*d^2/dx^2 P(n-3,x) - (147*n^2+1855*n-2748+(4116*n^2-24794*n+27930)*x+(1715*n^2-9800*n+13608)*x^2+(5488*n-10192)*x^3)*P(n-2,x) - (147*n^2-497*n+388+(2058*n-8918)*x-(1029*n^2-2450*n-8456)*x^2+1372*x^3-5488*x^4)*d/dx P(n-2,x) + (2744*x^2-5488*x^3+2744*x^4)*d^2/dx^2 P(n-2,x) + (294*n+56+(784*n-2828)*x+784*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 24 + 25*x + x^2, P(3,x) = 207 + 618*x + 209*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 24 + 25*x + x^2;
P[3, x_] := 207 + 618*x + 209*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((2058*n^2 - 6958*n + 5432 + (1029*n^2 - 3479*n - 1400)*x - 4802*x^2 - (7203*n^2 - 31556*n + 29988)*x^3)*P[n - 4, x] + ((1029*n^2 - 3479*n + 2716)*x - 2058*x^2 - (7203*n^2 - 31556*n + 29988)*x^3)*D[P[n - 4, x], x] + (2058*n^3 - 14749*n^2 + 31773*n - 20564 + (1029*n^3 - 4361*n^2 - 2534*n + 24230)*x - (1029*n^2 + 3038*n - 18648)*x^2 - (7203*n^3 - 52136*n^2 + 120148*n - 80192)*x^3)*P[n - 3, x] + (2205*n^2 - 7455*n + 5820 + (1029*n^3 - 9506*n^2 + 23093*n - 20318)*x - (2058*n^2 - 5929*n - 5054)*x^2 - (7203*n^3 - 59339*n^2 + 151704*n - 117726)*x^3 + (7203*n^2 - 31556*n + 29988)*x^4)*D[P[n - 3, x], x] + ((1029*n^2 - 3479*n + 2716)*x - (1029*n^2 - 3479*n + 4774)*x^2 - (7203*n^2 - 31556*n + 27930)*x^3 + (7203*n^2 - 31556*n + 29988)*x^4)*D[P[n - 3, x], {x, 2}] - (147*n^2 + 1855*n - 2748 + (4116*n^2 - 24794*n + 27930)*x + (1715*n^2 - 9800*n + 13608)*x^2 + (5488*n - 10192)*x^3)*P[n - 2, x] - (147*n^2 - 497*n + 388 + (2058*n - 8918)*x - (1029*n^2 - 2450*n - 8456)*x^2 + 1372*x^3 - 5488*x^4)*D[P[n - 2, x], x] + (2744*x^2 - 5488*x^3 + 2744*x^4)*D[P[n - 2, x], {x, 2}] + (294*n + 56 + (784*n - 2828)*x + 784*x^2)*P[n - 1, x])/(56)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 24 + 25*x + x^2; v[4] = 207 + 618*x + 209*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2058*n^2 - 6958*n + 5432 + (1029*n^2 - 3479*n - 1400)*x - 4802*x^2 - (7203*n^2 - 31556*n + 29988)*x^3)*v[i - 4] + ((1029*n^2 - 3479*n + 2716)*x - 2058*x^2 - (7203*n^2 - 31556*n + 29988)*x^3)*deriv(v[i - 4], x) + (2058*n^3 - 14749*n^2 + 31773*n - 20564 + (1029*n^3 - 4361*n^2 - 2534*n + 24230)*x - (1029*n^2 + 3038*n - 18648)*x^2 - (7203*n^3 - 52136*n^2 + 120148*n - 80192)*x^3)*v[i - 3] + (2205*n^2 - 7455*n + 5820 + (1029*n^3 - 9506*n^2 + 23093*n - 20318)*x - (2058*n^2 - 5929*n - 5054)*x^2 - (7203*n^3 - 59339*n^2 + 151704*n - 117726)*x^3 + (7203*n^2 - 31556*n + 29988)*x^4)*deriv(v[i - 3], x) + ((1029*n^2 - 3479*n + 2716)*x - (1029*n^2 - 3479*n + 4774)*x^2 - (7203*n^2 - 31556*n + 27930)*x^3 + (7203*n^2 - 31556*n + 29988)*x^4)*deriv(deriv(v[i - 3], x), x) - (147*n^2 + 1855*n - 2748 + (4116*n^2 - 24794*n + 27930)*x + (1715*n^2 - 9800*n + 13608)*x^2 + (5488*n - 10192)*x^3)*v[i - 2] - (147*n^2 - 497*n + 388 + (2058*n - 8918)*x - (1029*n^2 - 2450*n - 8456)*x^2 + 1372*x^3 - 5488*x^4)*deriv(v[i - 2], x) + (2744*x^2 - 5488*x^3 + 2744*x^4)*deriv(deriv(v[i - 2], x), x) + (294*n + 56 + (784*n - 2828)*x + 784*x^2)*v[i - 1])/(56)); 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

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.