Real-rooted atlas

derivative of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.

nonnegative coefficients

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

Recurrence

\(\left(1 - 3\,x + 2\,n x\right)\,P_{n}(x) = \left(-2 - 2\,x + 6\,x^{2} + 14\,x^{3} - 28\,x^{4} + 12\,x^{5} - 4\,n x - 10\,n x^{2} - 2\,n x^{3} + 54\,n x^{4} - 38\,n x^{5} - 4\,n^{2} x^{3} - 20\,n^{2} x^{4} + 32\,n^{2} x^{5} - 8\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-x - x^{2} + 6\,x^{3} - 2\,x^{4} - 5\,x^{5} + 3\,x^{6} - 2\,n x^{2} - 6\,n x^{3} + 10\,n x^{4} + 6\,n x^{5} - 8\,n x^{6} - 4\,n^{2} x^{4} + 4\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(3 - 3\,x - 15\,x^{2} + 15\,x^{3} + 6\,n x + 12\,n x^{2} - 22\,n x^{3} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x - 3\,x^{2} - x^{3} + 3\,x^{4} + 2\,n x^{2} - 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27Pn(1)
00
1224
262018448
31494204188706576
430352135024242250105621087680
5621194717620648330603097216744477655810115200
6126385234146143868345660512952483924287736102438192601386121935360
7254120861526768951403002730636246689396728483288540491422378505598127251633021436126720
8510373046576305197072233077506651362412840135017313272016508745011085604051277050155912162849730261128765016743178240
9102211398627669122885816016818388062287687215739794562827216432369129762035340205402473396288124575374443727808810100356013834560911888173741816721510400
1020463460841146479415554475611550330005413088016173416190643983914766467558467636859513354808257146044459758721496322058639761263053870434650990006221790246496716631147563887420408748032000
1140941046478470201888214477567663003710446300865901772362304165064520757281077371901756174298076790021708079887122091576921480155620385809288629113252437979192232011901356424026054212614369651490229779452410464780859742210790948044800
1281903155856191490390427657802449598981250354213076080171513440553059747953614001555923864222031062924648480483192838728605911730829748857104608213380434880945078722593206440370011949590722800416532641343010626392282401884761287502138289012013404327303471441618837024306070526361600
131638295003787760188082203870053631530664686027332412953161593634389676066666863392600208728786513330503232658434630952439396371056143308154362622417270165312858641671928467563928129878099505144080517225349540839426548563629015000044263335043255790576920911080431772013242879168121212128529485949477528114004536409550269284139299635200
1432766285667163133854162112737905068197797738482020649760188168143579758919772712958455709128264708000796905075827389961625801712841790830211830974055891289668452930045805722165387758194045662452261159131429480412987411883862242647119131283054613648930193092644558828001682355017823961392728204307392767593167571578735661690986959701726764274304082611545135037136730888468228299949115834368000
Technical details
Node id
n-95696ed532f4af38 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-435ee979f8f45497

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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 ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    0;
  1:    2,      2;
  2:    6,     20,      18,        4;
  3:   14,     94,     204,      188,        70,         6;
  4:   30,    352,    1350,     2424,      2250,      1056,        210,          8;
  5:   62,   1194,    7176,    20648,     33060,     30972,      16744,       4776,        558,         10;
  6:  126,   3852,   34146,   143868,    345660,    512952,     483924,     287736,     102438,      19260,       1386,         12;
  7:  254,  12086,  152676,   895140,   3002730,   6362466,    8939672,    8483288,    5404914,    2237850,     559812,      72516,      3302,        14;
  8:  510,  37304,  657630,  5197072,  23307750,  66513624,  128401350,  173132720,  165087450,  110856040,   51277050,   15591216,   2849730,    261128,     7650,     16;
  9: 1022, 113986, 2766912, 28858160, 168183880, 622876872, 1573979456, 2827216432, 3691297620, 3534020540, 2473396288, 1245753744, 437278088, 101003560, 13834560, 911888, 17374, 18;
%F Formula
The row polynomials satisfy (1+(2*n-3)*x)*P(n,x) = -(2+(4*n+2)*x+(10*n-6)*x^2+(4*n^2+2*n-14)*x^3+(20*n^2-54*n+28)*x^4+(8*n^3-32*n^2+38*n-12)*x^5)*P(n-2,x) - (x+(2*n+1)*x^2+(6*n-6)*x^3+(4*n^2-10*n+2)*x^4-(6*n-5)*x^5-(4*n^2-8*n+3)*x^6)*d/dx P(n-2,x) + (3+(6*n-3)*x+(12*n-15)*x^2+(8*n^2-22*n+15)*x^3)*P(n-1,x) + (x+(2*n-3)*x^2-x^3-(2*n-3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 + (4*n + 2)*x + (10*n - 6)*x^2 + (4*n^2 + 2*n - 14)*x^3 + (20*n^2 - 54*n + 28)*x^4 + (8*n^3 - 32*n^2 + 38*n - 12)*x^5)*P[n - 2, x] - (x + (2*n + 1)*x^2 + (6*n - 6)*x^3 + (4*n^2 - 10*n + 2)*x^4 - (6*n - 5)*x^5 - (4*n^2 - 8*n + 3)*x^6)*D[P[n - 2, x], x] + (3 + (6*n - 3)*x + (12*n - 15)*x^2 + (8*n^2 - 22*n + 15)*x^3)*P[n - 1, x] + (x + (2*n - 3)*x^2 - x^3 - (2*n - 3)*x^4)*D[P[n - 1, x], x])/(1 + (2*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2 + (4*n + 2)*x + (10*n - 6)*x^2 + (4*n^2 + 2*n - 14)*x^3 + (20*n^2 - 54*n + 28)*x^4 + (8*n^3 - 32*n^2 + 38*n - 12)*x^5)*v[i - 2] - (x + (2*n + 1)*x^2 + (6*n - 6)*x^3 + (4*n^2 - 10*n + 2)*x^4 - (6*n - 5)*x^5 - (4*n^2 - 8*n + 3)*x^6)*deriv(v[i - 2], x) + (3 + (6*n - 3)*x + (12*n - 15)*x^2 + (8*n^2 - 22*n + 15)*x^3)*v[i - 1] + (x + (2*n - 3)*x^2 - x^3 - (2*n - 3)*x^4)*deriv(v[i - 1], x))/(1 + (2*n - 3)*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

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.