Real-rooted atlas

A038338 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 5\,x + 4\,x^{2} + 3\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(12 + 108\,x + 204\,x^{2} + 108\,x^{3} + 36\,n x + 108\,n x^{2} + 72\,n x^{3} + 12\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 1152\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0144115212962592
1172820736466562764896768
220736331776111974413271045184003317760
324883249766402239488039813120311040008957952107495424
429859847166361640310784095551488011197440006449725441463132163344302080
53583180810032906246772211712200658124803135283200027088846848122903101442293235712100902371328
642998169613759414272108355387392385263599616752467968000866843099136589934886912220150628352348285173762972033482752
751597803521857520926721671768834048693474479308816253308108800234047636766722123765592883211888133931008376147987660851597803520085858745057280

All 15 rows

Technical details
Node id
n-59163859c5477b68 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:          144,           1152,            1296;
  1:         1728,          20736,           46656,            27648;
  2:        20736,         331776,         1119744,          1327104,           518400;
  3:       248832,        4976640,        22394880,         39813120,         31104000,           8957952;
  4:      2985984,       71663616,       403107840,        955514880,       1119744000,         644972544,         146313216;
  5:     35831808,     1003290624,      6772211712,      20065812480,      31352832000,       27088846848,       12290310144,        2293235712;
  6:    429981696,    13759414272,    108355387392,     385263599616,     752467968000,      866843099136,      589934886912,      220150628352,      34828517376;
  7:   5159780352,   185752092672,   1671768834048,    6934744793088,   16253308108800,    23404763676672,    21237655928832,    11888133931008,    3761479876608,     515978035200;
  8:  61917364224,  2476694568960,  25076532510720,  118881339310080,  325066162176000,   561714328240128,   637129677864960,   475525357240320,  225688792596480,   61917364224000,   7492001071104;
  9: 743008370688, 32692368310272, 367789143490560, 1961542098616320, 6129819058176000, 12357715221282816, 16820223495634944, 15692336788930560, 9930306874245120, 4086546038784000, 988944141385728, 106993205379072;
%F Formula
The row polynomials satisfy (1+(3*n+5)*x+(n^2+4*n+4)*x^2)*P(n,x) = (12+(36*n+108)*x+(12*n^2+108*n+204)*x^2+(12*n^2+72*n+108)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 1152*x + 1296*x^2.
%t Mathematica
P[0, x_] := 144 + 1152*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + (36*n + 108)*x + (12*n^2 + 108*n + 204)*x^2 + (12*n^2 + 72*n + 108)*x^3)*P[n - 1, x])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 1152*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + (36*n + 108)*x + (12*n^2 + 108*n + 204)*x^2 + (12*n^2 + 72*n + 108)*x^3)*v[i - 1])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*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.