Real-rooted atlas

A272866 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 25\,x - 30\,x^{2} - 5\,x^{3} + x^{4} + 75\,n x + 40\,n x^{2} - 5\,n x^{3} - 4\,n x^{4} + 25\,n^{2} x^{2} + 20\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(125 + 375\,x + 450\,x^{2} + 275\,x^{3} + 90\,x^{4} + 15\,x^{5} + x^{6} + 375\,n x + 825\,n x^{2} + 700\,n x^{3} + 285\,n x^{4} + 55\,n x^{5} + 4\,n x^{6} + 125\,n^{2} x^{2} + 225\,n^{2} x^{3} + 145\,n^{2} x^{4} + 40\,n^{2} x^{5} + 4\,n^{2} x^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1520934
22520031516025725
312515004050440022505404912914
462510000382506400056875288008330128081208241
53125625003093757000008906256975003491251120002227525001213149146
61562537500022781256500000107812501147500082381254080000139725032500049005432016945518869
778125218750015750000542500001110156251496250001404156259460000046423125166250004298525781200946406860225636151450
8390625125000001040625004200000001022656250166950000019379500001656000000106287187551750000019142200053424000110610501646400166500102402898661161729
91953125703125006644531253075000000868359375016655625000230560312502390400000018997284375117359375005675731875215136000063615825014508900025008750314880027310514580361115480975346
10976562539062500041308593752156250000069287109375152718750000245650781250299700000000284366953125213296875000127759740625614295000002373235312573426500001804781250347520000512613755589000424175200004411513588058741
11488281252148437500251367187501460937500005263671875001310976562500240339257812533742500000003724449609375329011718750023540398750001374518250000657574775000257945187500827655468752159520000045360356257551225009728950093500006306302662052919556818149154
12244140625117187500001502929687509625000000003843017578125106734375000002197056640625034897500000000439478789062504471328125000037239092343750256178700000001464149720312569737535000002769849843750915728000000250882706250565339500001035934625015180000001735445251490720090459034560625249647711784625
1312207031256347656250088549804687561953125000002715209960937583202539062500190023818359375336943750000000476893828125000549376953125000522361348437500414064462500000275546195234375154640891562500733702886718752944458400000099809870875002847970125000680133025000134777500000218907438752863646500294163675228384001259375439407293153835206933994
14610351562534179687500051525878906253910156250000018662109375000062544726562500015695767089843753073606250000000483081231445312562171484375000006647934261718750596969100000000045370633191406252934634040625000162131004492187576659828000000031034586964062510745443800000031734271318750795795000000016832697131252975250850004337925637551236640004777500003383380017095055488084139484567407526901
Technical details
Node id
n-f0ea69ccf00996a4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-12cf750f588dcd1a

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:       5,       20,         9;
  2:      25,      200,       315,        160,         25;
  3:     125,     1500,      4050,       4400,       2250,         540,          49;
  4:     625,    10000,     38250,      64000,      56875,       28800,        8330,        1280,          81;
  5:    3125,    62500,    309375,     700000,     890625,      697500,      349125,      112000,       22275,        2500,        121;
  6:   15625,   375000,   2278125,    6500000,   10781250,    11475000,     8238125,     4080000,     1397250,      325000,      49005,       4320,       169;
  7:   78125,  2187500,  15750000,   54250000,  111015625,   149625000,   140415625,    94600000,    46423125,    16625000,    4298525,     781200,     94640,      6860,      225;
  8:  390625, 12500000, 104062500,  420000000, 1022656250,  1669500000,  1937950000,  1656000000,  1062871875,   517500000,  191422000,   53424000,  11061050,   1646400,   166500,   10240,    289;
  9: 1953125, 70312500, 664453125, 3075000000, 8683593750, 16655625000, 23056031250, 23904000000, 18997284375, 11735937500, 5675731875, 2151360000, 636158250, 145089000, 25008750, 3148800, 273105, 14580, 361;
%F Formula
The row polynomials satisfy (25+(75*n-25)*x+(25*n^2+40*n-30)*x^2+(20*n^2-5*n-5)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (125+(375*n+375)*x+(125*n^2+825*n+450)*x^2+(225*n^2+700*n+275)*x^3+(145*n^2+285*n+90)*x^4+(40*n^2+55*n+15)*x^5+(4*n^2+4*n+1)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((125 + (375*n + 375)*x + (125*n^2 + 825*n + 450)*x^2 + (225*n^2 + 700*n + 275)*x^3 + (145*n^2 + 285*n + 90)*x^4 + (40*n^2 + 55*n + 15)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(25 + (75*n - 25)*x + (25*n^2 + 40*n - 30)*x^2 + (20*n^2 - 5*n - 5)*x^3 + (4*n^2 - 4*n + 1)*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; for(n = 1, N, my(i = n + 1); v[i] = ((125 + (375*n + 375)*x + (125*n^2 + 825*n + 450)*x^2 + (225*n^2 + 700*n + 275)*x^3 + (145*n^2 + 285*n + 90)*x^4 + (40*n^2 + 55*n + 15)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(25 + (75*n - 25)*x + (25*n^2 + 40*n - 30)*x^2 + (20*n^2 - 5*n - 5)*x^3 + (4*n^2 - 4*n + 1)*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

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