Real-rooted atlas

A038301 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(44 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1936 + 1760\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0193617604004096
185184116160528008000262144
2374809668147204646400140800016000016777216
31649162243748096003407360001548800003520000032000001073741824
4725631385619789946880224885760001362944000046464000008448000006400000068719476736
53192778096641015883939840138529628160010494668800004770304000001300992000001971200000012800000004398046511104
6140482236252165108444954624081270715187200738824683520004197867520000015264972800000346931200000045056000000025600000000281474976710656
761812183950950425286802525388804597600459161600487624291123200033247110758400001511232307200000457949184000000892108800000001013760000000051200000000018014398509481984
82719736093841817612362436790190080025286802525388800030650669727744000024381214556160000013298844303360000050374410240000000130842624000000002230272000000000225280000000000102400000000001152921504606846976
911966838812903997445983419406451998720135986804692090880001854365518528512000016857868350259200000107277344047104000004876242911232000000158319575040000000035981721600000000054517760000000000495616000000000020480000000000073786976294838206464
105265409077677758873628720413150969593856071801032877423984640010878944375367270400001112619311117107200000809177680812441600000429109376188416000000167185471242240000000474958725120000000009595125760000000000130842624000000000010813440000000000040960000000000004722366482869645213696
112316779994178213904384136900636019621730713603733653709626047201280062227561827100786688000707131384398872576000005785620417808957440000035064366168539136000000159383482584268800000005433527815372800000000137210298368000000000024947326976000000000030926438400000000000234291200000000000081920000000000000302231454903657293676544
1210193831974384141179289664869839836989989322752019166089042747042299904003484743462317644054528000435592932789705506816000039599357526336864256000002699956194977513472000000140257464674156544000000055784218904494080000000016904308758937600000000038418883543040000000000635022868480000000000072161689600000000000050462720000000000000163840000000000000019342813113834066795298816
1344852860687290221188874243058149592315242353786880097304759755484983984128000191660890427470422999040000261355759673823304089600000261355759673823304089600000197996787631684321280000000115712408356179148800000000525965492528087040000000001859473963483136000000000050712926276812800000000001047787732992000000000000158755717120000000000000166526976000000000000001081344000000000000000327680000000000000001237940039285380274899124224
1419735258702407697323104665614352915419932870780439756804893039347704387766059008000103791743739183982916403200001533287123419763383992320000016726768619124691461734400000139389738492705762181120000009051281720305568972800000000462849633424716595200000000018700995289887539200000000005950316683146035200000000001475285128052736000000000002794100621312000000000000039078330368000000000000003806330880000000000000002306867200000000000000065536000000000000000079228162514264337593543950336
Technical details
Node id
n-1b95b22bf8059e31 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-2235b7dd30e0ad6a

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:                1936,                1760,                  400;
  1:               85184,              116160,                52800,                 8000;
  2:             3748096,             6814720,              4646400,              1408000,               160000;
  3:           164916224,           374809600,            340736000,            154880000,             35200000,              3200000;
  4:          7256313856,         19789946880,          22488576000,          13629440000,           4646400000,            844800000,            64000000;
  5:        319277809664,       1015883939840,        1385296281600,        1049466880000,         477030400000,         130099200000,         19712000000,          1280000000;
  6:      14048223625216,      51084449546240,       81270715187200,       73882468352000,       41978675200000,       15264972800000,       3469312000000,        450560000000,        25600000000;
  7:     618121839509504,    2528680252538880,     4597600459161600,     4876242911232000,     3324711075840000,     1511232307200000,     457949184000000,      89210880000000,     10137600000000,      512000000000;
  8:   27197360938418176,  123624367901900800,   252868025253888000,   306506697277440000,   243812145561600000,   132988443033600000,   50374410240000000,   13084262400000000,   2230272000000000,   225280000000000,   10240000000000;
  9: 1196683881290399744, 5983419406451998720, 13598680469209088000, 18543655185285120000, 16857868350259200000, 10727734404710400000, 4876242911232000000, 1583195750400000000, 359817216000000000, 54517760000000000, 4956160000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = (44+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1936 + 1760*x + 400*x^2.
%t Mathematica
P[0, x_] := 1936 + 1760*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(44 + 20*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] = 1936 + 1760*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (44 + 20*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.