Real-rooted atlas

A038319 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(21 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 441 + 420\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0441420100961
19261132306300100029791
21944813704402646008400010000923521
34084101972405092610004410000105000010000028629151
48576612124504606029172150018522000066150000126000001000000887503681
5180108854160036284708576612100680683500032413500009261000001470000001000000027512614111
63782285936114408708328024014513880022870965600013613670000051861600000123480000001680000000100000000852891037441
77942800465813404057342490648391874760072043541640005145967260000245046060000077792400000015876000000018900000000100000000026439622160671
816679880978201794280046581001702028671245002161306249200001801088541000001029193452000004084101000000011113200000000198450000000021000000000010000000000819628286980801
935027750054222118347869076021104368540256195500624077179456500059435921853000003962394790200000188685466200000064178730000000015280650000000024255000000000231000000000010000000000025408476896404831
10735582751138664142033300065066520110087214456126600174741610247820000187223153836950000142646212447200000792478958040000003234607992000000096268095000000002037420000000000291060000000000252000000000001000000000000787662783788549761
111544723777391194619562575764802633302732164504229323800477044595976548600056791023330541500004867801999760700000309066793635600000014717466363600000005256237987000000001390539150000000002648646000000000034398000000000002730000000000001000000000000024417546297445042591
1232439199325215086812162613288347672454066938030353618433100127501010197368444000166965608591792010000159014865325516200000113582046661083000000618133587271200000002575556613630000000081763702020000000001946754810000000000337100400000000000401310000000000002940000000000000100000000000000756943935220796320321
1368122318582951682301486587989878226302150162195996626075434050033469015176809216550004781287882401316650000500896825775376030000039753716331379050000002433900999880350000000115900047613350000000042925943560500000000012264555303000000000026546656500000000000421375500000000000046305000000000000031500000000000000100000000000000023465261991844685929951
1414305686902419853283211089957097327226916816038927039190258104172000865045315339068981600001338760607072368662000001530012122368421328000001335724868734336080000009086563732886640000000048678019997607000000000206044529090400000000006868150969680000000000178393531680000000000035395542000000000000051861600000000000000529200000000000000033600000000000000010000000000000000727423121747185263828481
Technical details
Node id
n-582aac7b79b62cfa · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-09bb967aa8554943

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             441,              420,              100;
  1:            9261,            13230,             6300,             1000;
  2:          194481,           370440,           264600,            84000,            10000;
  3:         4084101,          9724050,          9261000,          4410000,          1050000,           100000;
  4:        85766121,        245046060,        291721500,        185220000,         66150000,         12600000,          1000000;
  5:      1801088541,       6003628470,       8576612100,       6806835000,       3241350000,        926100000,        147000000,        10000000;
  6:     37822859361,     144087083280,     240145138800,     228709656000,     136136700000,      51861600000,      12348000000,      1680000000,       100000000;
  7:    794280046581,    3404057342490,    6483918747600,    7204354164000,    5145967260000,    2450460600000,     777924000000,    158760000000,     18900000000,     1000000000;
  8:  16679880978201,   79428004658100,  170202867124500,  216130624920000,  180108854100000,  102919345200000,   40841010000000,  11113200000000,   1984500000000,   210000000000,   10000000000;
  9: 350277500542221, 1834786907602110, 4368540256195500, 6240771794565000, 5943592185300000, 3962394790200000, 1886854662000000, 641787300000000, 152806500000000, 24255000000000, 2310000000000, 100000000000;
%F Formula
The row polynomials satisfy P(n,x) = (21+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 420*x + 100*x^2.
%t Mathematica
P[0, x_] := 441 + 420*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 441 + 420*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 10*x)*v[i - 1]); 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.