Real-rooted atlas

theta_plus_one 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

\(\left(1 - 2\,x + 2\,n x\right)\,P_{n}(x) = \left(-6\,x^{2} + 12\,x^{3} - 6\,x^{4} + 2\,n x^{2} - 20\,n x^{3} + 34\,n x^{4} - 12\,n x^{5} + 4\,n^{2} x^{3} - 20\,n^{2} x^{4} + 20\,n^{2} x^{5} - 8\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 2\,x^{3} + 2\,x^{4} - 2\,x^{5} - 6\,n x^{3} + 4\,n x^{4} + 6\,n x^{5} - 4\,n x^{6} - 4\,n^{2} x^{4} + 4\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(1 + 2\,x - 9\,x^{2} + 6\,x^{3} + 2\,n x + 12\,n x^{2} - 14\,n x^{3} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x - 2\,x^{2} - x^{3} + 2\,x^{4} + 2\,n x^{2} - 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11438
21123024572
3128141272235847768
4160528180030302700123224099600
511241791956825810396723613419136537362011138240
61252577845528179835414792598444553056323703113820211861512132257920
7150818129203568111892536032767422877102167689543699600546024616356107047855935561541287680
811020559568768406496340279693007759922814674440019477431018343050012194164455938600168904843068940279780816017836075520
91204417097936892163607270020182065672668968417988336643180618486410144180038874225942698250496134956655647091486410821810014756864968881183961918579456000
10140925191261528639219443094513860396006315269352198189932164481904112275064964040945464690289007795684864738614954346832381281353272004036961056006610652136878876432877984092021449622835200
111818815697176269358410268096959195604452520684343552025556919045697585851941197079890840191727884469023681541695042265874998270167591184717694959764199040511138380812645191200527586813356390187684131346792010988019900682311771943321600
121163804733784255320520534572253059518777500413248588760196015360632067216447815751728804293580034169217113328527119460431206404375065561261497270383640465943869727202766086869728012696440142975441034561422011216747409201983959250002245203462614042628603629234419656025331576403558400
1313276414250567103469174427548375670378367976232318878151120218212964453440750002213166752319208739037005535559242780931039024796041152155250500559507618598639567693921791351929532780138536639472153685549551933887141745757302666015833380055742545532411133600956634453360613873492461841267225280826208150464118754725425932279998303861145600
14165532428500744178472216140922381335237357286178424091386886196164091153051168802078262672769294120000885450083410128957788381868554680905685633555227215563807487770818560008485772598065048924055745236406858112438799125126603632802832021358528414407204092708902588240001770900018715159462364614512506708907127915741405464494942914471367046119066752716006940438565066691744829321374052679680000
Technical details
Node id
n-7b7663b683c07e49 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-880bd7c6b138382f

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,    4,      3;
  2: 1,   12,     30,      24,        5;
  3: 1,   28,    141,     272,      235,        84,         7;
  4: 1,   60,    528,    1800,     3030,      2700,      1232,        240,          9;
  5: 1,  124,   1791,    9568,    25810,     39672,     36134,      19136,       5373,        620,         11;
  6: 1,  252,   5778,   45528,   179835,    414792,    598444,     553056,     323703,     113820,      21186,       1512,         13;
  7: 1,  508,  18129,  203568,  1118925,   3603276,   7422877,   10216768,    9543699,    6005460,    2461635,     610704,      78559,      3556,        15;
  8: 1, 1020,  55956,  876840,  6496340,  27969300,  77599228,  146744400,  194774310,  183430500,  121941644,   55938600,   16890484,   3068940,    279780,     8160,     17;
  9: 1, 2044, 170979, 3689216, 36072700, 201820656, 726689684, 1798833664, 3180618486, 4101441800, 3887422594, 2698250496, 1349566556, 470914864, 108218100, 14756864, 968881, 18396, 19;
%F Formula
The row polynomials satisfy (1+(2*n-2)*x)*P(n,x) = ((2*n-6)*x^2+(4*n^2-20*n+12)*x^3-(20*n^2-34*n+6)*x^4-(8*n^3-20*n^2+12*n)*x^5)*P(n-2,x) - (2*x^2+(6*n-2)*x^3+(4*n^2-4*n-2)*x^4-(6*n-2)*x^5-(4*n^2-4*n)*x^6)*d/dx P(n-2,x) + (1+(2*n+2)*x+(12*n-9)*x^2+(8*n^2-14*n+6)*x^3)*P(n-1,x) + (x+(2*n-2)*x^2-x^3-(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 6)*x^2 + (4*n^2 - 20*n + 12)*x^3 - (20*n^2 - 34*n + 6)*x^4 - (8*n^3 - 20*n^2 + 12*n)*x^5)*P[n - 2, x] - (2*x^2 + (6*n - 2)*x^3 + (4*n^2 - 4*n - 2)*x^4 - (6*n - 2)*x^5 - (4*n^2 - 4*n)*x^6)*D[P[n - 2, x], x] + (1 + (2*n + 2)*x + (12*n - 9)*x^2 + (8*n^2 - 14*n + 6)*x^3)*P[n - 1, x] + (x + (2*n - 2)*x^2 - x^3 - (2*n - 2)*x^4)*D[P[n - 1, x], x])/(1 + (2*n - 2)*x)]];
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] = 1 + 4*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 6)*x^2 + (4*n^2 - 20*n + 12)*x^3 - (20*n^2 - 34*n + 6)*x^4 - (8*n^3 - 20*n^2 + 12*n)*x^5)*v[i - 2] - (2*x^2 + (6*n - 2)*x^3 + (4*n^2 - 4*n - 2)*x^4 - (6*n - 2)*x^5 - (4*n^2 - 4*n)*x^6)*deriv(v[i - 2], x) + (1 + (2*n + 2)*x + (12*n - 9)*x^2 + (8*n^2 - 14*n + 6)*x^3)*v[i - 1] + (x + (2*n - 2)*x^2 - x^3 - (2*n - 2)*x^4)*deriv(v[i - 1], x))/(1 + (2*n - 2)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.