Real-rooted atlas

A376827 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - 5\,n x + 5\,n x^{2} - n x^{3} + 6\,n^{2} x - 17\,n^{2} x^{2} + 5\,n^{2} x^{3} + 11\,n^{3} x^{2} - 11\,n^{3} x^{3} + n^{3} x^{4} + 6\,n^{4} x^{3} - 2\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-3 - 9\,x - 6\,x^{2} + 6\,x^{3} + 9\,x^{4} + 3\,x^{5} + 3\,n - 9\,n x - 45\,n x^{2} - 57\,n x^{3} - 30\,n x^{4} - 6\,n x^{5} + 18\,n^{2} x + 18\,n^{2} x^{2} - 36\,n^{2} x^{3} - 57\,n^{2} x^{4} - 24\,n^{2} x^{5} - 3\,n^{2} x^{6} + 33\,n^{3} x^{2} + 69\,n^{3} x^{3} + 36\,n^{3} x^{4} - 3\,n^{3} x^{5} - 3\,n^{3} x^{6} + 18\,n^{4} x^{3} + 39\,n^{4} x^{4} + 24\,n^{4} x^{5} + 3\,n^{4} x^{6} + 3\,n^{5} x^{4} + 6\,n^{5} x^{5} + 3\,n^{5} x^{6}\right)\,P_{n-2}(x) + \left(-1 + x + 8\,x^{2} + 8\,x^{3} + x^{4} - x^{5} + 2\,n - 8\,n x - 16\,n x^{2} - 4\,n x^{3} + 4\,n x^{4} + 2\,n x^{5} + 12\,n^{2} x - 11\,n^{2} x^{2} - 33\,n^{2} x^{3} - 8\,n^{2} x^{4} + 2\,n^{2} x^{5} + 22\,n^{3} x^{2} + 12\,n^{3} x^{3} - 14\,n^{3} x^{4} - 4\,n^{3} x^{5} + 12\,n^{4} x^{3} + 11\,n^{4} x^{4} - n^{4} x^{5} + 2\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
23242754
3784189112392
419304102612164753040
5511020459081606375183622032
61413384190354512052875304566909157920
73931100474277220080343875297108134799251521106688
8110735424278964991872193725022317121518804566784896677651584
9313911300410170364218816988785014238504129201247232256228833131390052232960
108953358120362596517189760470032508122161692126370687590403263368589530001083313352963072
1125653112873212698235677239202116372504266606965807326145417913603428523451410915003414414336940322364180480
1273789354187243830666259737280913138875210387196833408707643740216832295856995516233580005892789541275073921247034115716466688
132129411107293214948458297441801638063203759865982412179049310442338603238422198460427152252815007369036246239175331246783137741736436103792553984
14616227345087125046899133588906048154210806754441271234490675954369135353028096149892904161123368645400746380304673230015443294769550331690926888138651075681497763840
Technical details
Node id
n-9c8afa4b9d121b03 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:    1,      4;
  2:    3,     24,      27;
  3:    7,     84,     189,     112;
  4:   19,    304,    1026,    1216,     475;
  5:   51,   1020,    4590,    8160,    6375,     1836;
  6:  141,   3384,   19035,   45120,   52875,    30456,     6909;
  7:  393,  11004,   74277,  220080,  343875,   297108,   134799,   25152;
  8: 1107,  35424,  278964,  991872, 1937250,  2231712,  1518804,  566784,   89667;
  9: 3139, 113004, 1017036, 4218816, 9887850, 14238504, 12920124, 7232256, 2288331, 313900;
%F Formula
The row polynomials satisfy (n+(6*n^2-5*n)*x+(11*n^3-17*n^2+5*n)*x^2+(6*n^4-11*n^3+5*n^2-n)*x^3+(n^5-2*n^4+n^3)*x^4)*P(n,x) = (3*n-3+(18*n^2-9*n-9)*x+(33*n^3+18*n^2-45*n-6)*x^2+(18*n^4+69*n^3-36*n^2-57*n+6)*x^3+(3*n^5+39*n^4+36*n^3-57*n^2-30*n+9)*x^4+(6*n^5+24*n^4-3*n^3-24*n^2-6*n+3)*x^5+(3*n^5+3*n^4-3*n^3-3*n^2)*x^6)*P(n-2,x) + (2*n-1+(12*n^2-8*n+1)*x+(22*n^3-11*n^2-16*n+8)*x^2+(12*n^4+12*n^3-33*n^2-4*n+8)*x^3+(2*n^5+11*n^4-14*n^3-8*n^2+4*n+1)*x^4+(2*n^5-n^4-4*n^3+2*n^2+2*n-1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (18*n^2 - 9*n - 9)*x + (33*n^3 + 18*n^2 - 45*n - 6)*x^2 + (18*n^4 + 69*n^3 - 36*n^2 - 57*n + 6)*x^3 + (3*n^5 + 39*n^4 + 36*n^3 - 57*n^2 - 30*n + 9)*x^4 + (6*n^5 + 24*n^4 - 3*n^3 - 24*n^2 - 6*n + 3)*x^5 + (3*n^5 + 3*n^4 - 3*n^3 - 3*n^2)*x^6)*P[n - 2, x] + (2*n - 1 + (12*n^2 - 8*n + 1)*x + (22*n^3 - 11*n^2 - 16*n + 8)*x^2 + (12*n^4 + 12*n^3 - 33*n^2 - 4*n + 8)*x^3 + (2*n^5 + 11*n^4 - 14*n^3 - 8*n^2 + 4*n + 1)*x^4 + (2*n^5 - n^4 - 4*n^3 + 2*n^2 + 2*n - 1)*x^5)*P[n - 1, x])/(n + (6*n^2 - 5*n)*x + (11*n^3 - 17*n^2 + 5*n)*x^2 + (6*n^4 - 11*n^3 + 5*n^2 - n)*x^3 + (n^5 - 2*n^4 + n^3)*x^4)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (18*n^2 - 9*n - 9)*x + (33*n^3 + 18*n^2 - 45*n - 6)*x^2 + (18*n^4 + 69*n^3 - 36*n^2 - 57*n + 6)*x^3 + (3*n^5 + 39*n^4 + 36*n^3 - 57*n^2 - 30*n + 9)*x^4 + (6*n^5 + 24*n^4 - 3*n^3 - 24*n^2 - 6*n + 3)*x^5 + (3*n^5 + 3*n^4 - 3*n^3 - 3*n^2)*x^6)*v[i - 2] + (2*n - 1 + (12*n^2 - 8*n + 1)*x + (22*n^3 - 11*n^2 - 16*n + 8)*x^2 + (12*n^4 + 12*n^3 - 33*n^2 - 4*n + 8)*x^3 + (2*n^5 + 11*n^4 - 14*n^3 - 8*n^2 + 4*n + 1)*x^4 + (2*n^5 - n^4 - 4*n^3 + 2*n^2 + 2*n - 1)*x^5)*v[i - 1])/(n + (6*n^2 - 5*n)*x + (11*n^3 - 17*n^2 + 5*n)*x^2 + (6*n^4 - 11*n^3 + 5*n^2 - n)*x^3 + (n^5 - 2*n^4 + n^3)*x^4)); 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

Same fingerprint ?

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