Real-rooted atlas

A038336 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(44 + 20\,x + 22\,n + 10\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 440 + 200\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0440200640
11452013200300030720
2425920580800264000400001310720
3117128002129600014520000440000050000052428800
43092179207027680006388800002904000006600000060000002013265920
5793659328021645254400245968800001490720000050820000009240000007000000075161927680
619954863104063492746240086581017600065591680000029814400000081312000000123200000008000000002748779069440
749388286182401795937679360028571735808000259743052800001475812800000053665920000001219680000000158400000000900000000098956046499840
8120726921779200493882861824000897968839680000952391193600000649357632000000295162560000000894432000000001742400000000019800000000001000000000003518437208883200
92921591507056640132799613957120002716355740032000032925524121600000261907578240000001428586790400000054113136000000001405536000000000239580000000000242000000000001100000000000123848989752688640
1070118196169359360350590980846796800796797683742720000108654229601280000098776572364800000062857818777600000028571735808000000092765376000000000210830400000000003194400000000000290400000000000120000000000004323455642275676160
1116711503420363980809115365502016716800227884137550417920003452789962885120000035312624620416000000256819088148480000001361919406848000000053061795072000000001507437360000000000304532800000000000415272000000000003432000000000000130000000000000149879795598890106880
123959340810363158528023396104788509573120063807558514117017600010634593085686169600001208476487009792000000988753489371648000000599244539013120000000272383881369600000000928581413760000000002344902560000000000042634592000000000005285280000000000004004000000000000014000000000000005165088340638674452480
1393327319101417308160059390112155447377920001754707859138217984000031903779257058508800000398797240713231360000003625429461029376000000024718837234291200000000128409544074240000000005107197775680000000000154763568960000000000035173538400000000000058138080000000000000660660000000000000046200000000000000015000000000000000177088743107611695513600
142190081088246592831488014932371056226769305600047512089724357902336000093584419154038292480000012761511702823403520000001276151170282340352000000966781189607833600000000565001993926656000000000256819088148480000000000907946271232000000000002476217103360000000000051161510400000000000007751744000000000000008131200000000000000052800000000000000001600000000000000006044629098073145873530880
Technical details
Node id
n-5d1c1a2161cc3127 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2235b7dd30e0ad6a

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:              440,               200;
  1:            14520,             13200,              3000;
  2:           425920,            580800,            264000,             40000;
  3:         11712800,          21296000,          14520000,           4400000,            500000;
  4:        309217920,         702768000,         638880000,         290400000,          66000000,           6000000;
  5:       7936593280,       21645254400,       24596880000,       14907200000,        5082000000,         924000000,         70000000;
  6:     199548631040,      634927462400,      865810176000,      655916800000,      298144000000,       81312000000,      12320000000,        800000000;
  7:    4938828618240,    17959376793600,    28571735808000,    25974305280000,    14758128000000,     5366592000000,    1219680000000,     158400000000,      9000000000;
  8:  120726921779200,   493882861824000,   897968839680000,   952391193600000,   649357632000000,   295162560000000,   89443200000000,   17424000000000,   1980000000000,   100000000000;
  9: 2921591507056640, 13279961395712000, 27163557400320000, 32925524121600000, 26190757824000000, 14285867904000000, 5411313600000000, 1405536000000000, 239580000000000, 24200000000000, 1100000000000;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (22*n+44+(10*n+20)*x)*P(n-1,x) for n >= 1, with P(0,x) = 440 + 200*x.
%t Mathematica
P[0, x_] := 440 + 200*x;
P[n_, x_] := P[n, x] = Expand[Together[((22*n + 44 + (10*n + 20)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 440 + 200*x; for(n = 1, N, my(i = n + 1); v[i] = ((22*n + 44 + (10*n + 20)*x)*v[i - 1])/(n + 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.