Real-rooted atlas

A038329 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 50\,x + 16\,x^{2} + 30\,n x + 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(375 + 1350\,x + 1020\,x^{2} + 216\,x^{3} + 450\,n x + 540\,n x^{2} + 144\,n x^{3} + 60\,n^{2} x^{2} + 24\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 225 + 720\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02257203241269
133751620014580345637611
250625324000437400207360324001051785
37593756075000109350007776000243000027993628255311
411390625109350000246037500233280000109350000251942402286144736888509
5170859375191362500051667875006123600000382725000013226976002400451201791590418782780499
625628906253280500000010333575000014696640000011481750000052907904000144027072002149908480136048896470084109201
738443359375553584375000199290375000033067440000003100072500000178564176000064812182400014511882240018366600960100776960011590004761335

All 15 rows

Technical details
Node id
n-5ac21b852d5403a1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2e8ee919cea99b33

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:           225,             720,             324;
  1:          3375,           16200,           14580,             3456;
  2:         50625,          324000,          437400,           207360,            32400;
  3:        759375,         6075000,        10935000,          7776000,          2430000,           279936;
  4:      11390625,       109350000,       246037500,        233280000,        109350000,         25194240,         2286144;
  5:     170859375,      1913625000,      5166787500,       6123600000,       3827250000,       1322697600,       240045120,        17915904;
  6:    2562890625,     32805000000,    103335750000,     146966400000,     114817500000,      52907904000,     14402707200,      2149908480,      136048896;
  7:   38443359375,    553584375000,   1992903750000,    3306744000000,    3100072500000,    1785641760000,    648121824000,    145118822400,    18366600960,     1007769600;
  8:  576650390625,   9226406250000,  37366945312500,   70858800000000,   77501812500000,   53569252800000,  24304568400000,   7255941120000,  1377495072000,   151165440000,    7316407296;
  9: 8649755859375, 152235703125000, 685060664062500, 1461462750000000, 1826828437500000, 1473154452000000, 802050757200000, 299307571200000, 75762228960000, 12471148800000, 1207207203840, 52242776064;
%F Formula
The row polynomials satisfy (25+(30*n+50)*x+(4*n^2+16*n+16)*x^2)*P(n,x) = (375+(450*n+1350)*x+(60*n^2+540*n+1020)*x^2+(24*n^2+144*n+216)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 720*x + 324*x^2.
%t Mathematica
P[0, x_] := 225 + 720*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((375 + (450*n + 1350)*x + (60*n^2 + 540*n + 1020)*x^2 + (24*n^2 + 144*n + 216)*x^3)*P[n - 1, x])/(25 + (30*n + 50)*x + (4*n^2 + 16*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 225 + 720*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((375 + (450*n + 1350)*x + (60*n^2 + 540*n + 1020)*x^2 + (24*n^2 + 144*n + 216)*x^3)*v[i - 1])/(25 + (30*n + 50)*x + (4*n^2 + 16*n + 16)*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.