Real-rooted atlas

A038319 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 + 200\,x + 100\,x^{2} + 120\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1024 + 5760\,x + 6800\,x^{2} + 2250\,x^{3} + 1920\,n x + 3600\,n x^{2} + 1500\,n x^{3} + 400\,n^{2} x^{2} + 250\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 1280\,x + 900\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
025612809002436
1409630720432001600094016
265536655360138240010240002500003377296
31048576131072003686400040960000200000003600000115579776
4167772162516582408847360001310720000960000000345600000490000003818491456
52684354564697620480198180864003670016000035840000000193536000005488000000640000000122805902336
642949672968589934592042278584320093952409600011468800000008257536000003512320000008192000000081000000003866389852416
768719476736154618822656086973087744002254857830400033030144000000297271296000001685913600000058982400000001166400000000100000000000119641844381696
8109951162777627487790694400173946175488000515396075520000880803840000000951268147200000674365440000000314572800000000933120000000001600000000000012100000000003649461780530176
917592186044416483785116221440340161409843200011338713661440000221459251200000002790386565120000023737663488000000138412032000000005474304000000000140800000000000021296000000000014400000000000109980026521337856
10281474976710656844424930131968065310990689894400241892558110720000531502202880000000765363172147200000759605231616000000531502202880000000262766592000000000901120000000000002044416000000000027648000000000001690000000000003280158634601844736
1145035996273704961463669878895411201234971460318003200503136520870297600012283606466560000000198994424758272000002257112688230400000018425409699840000000109310902272000000004685824000000000000141746176000000000028753920000000000035152000000000000196000000000000096955819968269090816
127205759403792793625220157913274777602305280059260272640010245689152267878400027515278485094400000049527501273169920000063199155270451200000058961311039488000000040809403514880000000020992491520000000000079377858560000000000214695936000000000003937024000000000000439040000000000000225000000000000002843401192691482116096
13115292150460684697643234556422756761600425590164786511872000204913783045357568000060033334876569600000001188660030556078080000016853108072120320000000176883933118464000000001399179549081600000000083969966080000000000003810137210880000000000128817561600000000000031496192000000000000052684800000000000000540000000000000000025600000000000000082810958296047911960576
1418446744073709551616737869762948382064640778222015609621708800040352252661239644160000128071114403348480000000276633607111232716800000431439566646280192000000503136520870297600000000447737455706112000000000307090161664000000000000162565854330880000000000659545915392000000000002015756288000000000000044957696000000000000006912000000000000000006553600000000000000028900000000000000002396932620075708941664256
Technical details
Node id
n-31c6ec19afe4c4d5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-472634f5a19ab425

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:            256,            1280,              900;
  1:           4096,           30720,            43200,             16000;
  2:          65536,          655360,          1382400,           1024000,            250000;
  3:        1048576,        13107200,         36864000,          40960000,          20000000,           3600000;
  4:       16777216,       251658240,        884736000,        1310720000,         960000000,         345600000,          49000000;
  5:      268435456,      4697620480,      19818086400,       36700160000,       35840000000,       19353600000,        5488000000,         640000000;
  6:     4294967296,     85899345920,     422785843200,      939524096000,     1146880000000,      825753600000,      351232000000,       81920000000,       8100000000;
  7:    68719476736,   1546188226560,    8697308774400,    22548578304000,    33030144000000,    29727129600000,    16859136000000,     5898240000000,    1166400000000,     100000000000;
  8:  1099511627776,  27487790694400,  173946175488000,   515396075520000,   880803840000000,   951268147200000,   674365440000000,   314572800000000,   93312000000000,   16000000000000,   1210000000000;
  9: 17592186044416, 483785116221440, 3401614098432000, 11338713661440000, 22145925120000000, 27903865651200000, 23737663488000000, 13841203200000000, 5474304000000000, 1408000000000000, 212960000000000, 14400000000000;
%F Formula
The row polynomials satisfy (64+(120*n+200)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (1024+(1920*n+5760)*x+(400*n^2+3600*n+6800)*x^2+(250*n^2+1500*n+2250)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 1280*x + 900*x^2.
%t Mathematica
P[0, x_] := 256 + 1280*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1024 + (1920*n + 5760)*x + (400*n^2 + 3600*n + 6800)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*P[n - 1, x])/(64 + (120*n + 200)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 1280*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1024 + (1920*n + 5760)*x + (400*n^2 + 3600*n + 6800)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*v[i - 1])/(64 + (120*n + 200)*x + (25*n^2 + 100*n + 100)*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.