Real-rooted atlas

A038284 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 + 105\,x + 36\,x^{2} + 63\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(686 + 2646\,x + 2142\,x^{2} + 486\,x^{3} + 882\,n x + 1134\,n x^{2} + 324\,n x^{3} + 126\,n^{2} x^{2} + 54\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 672\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01966723241192
127441411213608345633920
23841626342438102419353632400908800
3537824460992088905606773760226800027993623360000
475295367744665618670176018966528095256000235146242286144582400000
5105413504126496204836593544964646799360311169600011522165762240421121791590414182400000
6147578905620239392768683079505921040883056648712748800043016085504125463582722006581248136048896338944000000
72066104678431877043609612295431106562185854418944219561269760013550066933765269470474241264146186241714216089610077696007976960000000

All 15 rows

Technical details
Node id
n-deb893c7b4f8d70a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fa8d58a7053642bf

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           196,            672,             324;
  1:          2744,          14112,           13608,            3456;
  2:         38416,         263424,          381024,          193536,            32400;
  3:        537824,        4609920,         8890560,         6773760,          2268000,          279936;
  4:       7529536,       77446656,       186701760,       189665280,         95256000,        23514624,         2286144;
  5:     105413504,     1264962048,      3659354496,      4646799360,       3111696000,      1152216576,       224042112,        17915904;
  6:    1475789056,    20239392768,     68307950592,    104088305664,      87127488000,     43016085504,     12546358272,      2006581248,      136048896;
  7:   20661046784,   318770436096,   1229543110656,   2185854418944,    2195612697600,   1355006693376,    526947047424,    126414618624,    17142160896,     1007769600;
  8:  289254654976,  4958651228160,  21517004436480,  43717088378880,   51230962944000,  37940187414528,  18443146659840,   5899348869120,  1199951262720,   141087744000,    7316407296;
  9: 4049565169664, 76363228913664, 368179853690880, 841553951293440, 1127081184768000, 973798143639552, 568048917123072, 227124931461120, 61597498152960, 10863756288000, 1126726723584, 52242776064;
%F Formula
The row polynomials satisfy (49+(63*n+105)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (686+(882*n+2646)*x+(126*n^2+1134*n+2142)*x^2+(54*n^2+324*n+486)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 672*x + 324*x^2.
%t Mathematica
P[0, x_] := 196 + 672*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((686 + (882*n + 2646)*x + (126*n^2 + 1134*n + 2142)*x^2 + (54*n^2 + 324*n + 486)*x^3)*P[n - 1, x])/(49 + (63*n + 105)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 196 + 672*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((686 + (882*n + 2646)*x + (126*n^2 + 1134*n + 2142)*x^2 + (54*n^2 + 324*n + 486)*x^3)*v[i - 1])/(49 + (63*n + 105)*x + (9*n^2 + 36*n + 36)*x^2)); 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.