Real-rooted atlas

A038301 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(40 + 22\,x + 20\,n + 11\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 440 + 242\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0440242682
11320014520399331713
2352000580800319440585641310804
388000001936000015972000585640080525550793655
421120000058080000063888000035138400096630600106293661889523966
549280000001626240000022360800000163979200006764142000148811124013641019768337783437
61126400000004336640000007155456000006559168000003607542400001190488992002182563152017148710482421110041768
7253440000000011151360000000214663680000002361300480000016233940800000714293395200019643068368003086767886402122152921984436212706659
8563200000000002787840000000006133248000000007871001600000006493576320000003571466976000001309537891200003086767886400042443058438002593742460102908358437673810
912390400000000006814720000000000168664320000000002473743360000000023809779840000000157144546944000007202458401600000226362978336000046687364281800057062334122200313842837672199175022724676921
1027033600000000000163553280000000000449771520000000000742123008000000000816335308800000000628578187776000000345718003276800000135817787001600000373498914254400006847480094664000753222810413040376611405206523353918950325437692
115857280000000000003865804800000000000116940595200000000002143910912000000000026530897536000000000233471898316800000001498111347532800000070625249240832000002427742942653600000593448274870880000979189653536952009791896535369520448795257871103112635778081762615823
121261568000000000000090202112000000000000297666969600000000000600295055360000000000825405701120000000000817151644108800000000599244539013120000000329584496457216000000135953604788601600000415413792409616000009139103433011552000137086551495173280012566267220390884053164976701653743760302129806536559014
132703360000000000000002081587200000000000000744167424000000000000016371683328000000000000247621710336000000000002723838813696000000000022471670212992000000000141250498481664000000006797680239430080000000249248275445769600000068543275747586640000013708655149517328000018849400830586326000159494930104961220062658722541234765124895749311431392852965
1457671680000000000000004757913600000000000000018317967360000000000000043657822208000000000000072035406643200000000000087162842038272000000000079899271868416000000000056500199392665600000000031075109665966080000000013293241357107712000000043867696478455449600000109669241196138624000002010602755262541440000255191888167937952000200507912131951248007351956778171545764129886110564664723671376
Technical details
Node id
n-49aa364e75170bd0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-590b4d5bb5323b01

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,              242;
  1:            13200,            14520,              3993;
  2:           352000,           580800,            319440,             58564;
  3:          8800000,         19360000,          15972000,           5856400,            805255;
  4:        211200000,        580800000,         638880000,         351384000,          96630600,          10629366;
  5:       4928000000,      16262400000,       22360800000,       16397920000,        6764142000,        1488111240,        136410197;
  6:     112640000000,     433664000000,      715545600000,      655916800000,      360754240000,      119048899200,      21825631520,       1714871048;
  7:    2534400000000,   11151360000000,    21466368000000,    23613004800000,    16233940800000,     7142933952000,    1964306836800,     308676788640,     21221529219;
  8:   56320000000000,  278784000000000,   613324800000000,   787100160000000,   649357632000000,   357146697600000,  130953789120000,   30867678864000,   4244305843800,   259374246010;
  9: 1239040000000000, 6814720000000000, 16866432000000000, 24737433600000000, 23809779840000000, 15714454694400000, 7202458401600000, 2263629783360000, 466873642818000, 57062334122200, 3138428376721;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (20*n+40+(11*n+22)*x)*P(n-1,x) for n >= 1, with P(0,x) = 440 + 242*x.
%t Mathematica
P[0, x_] := 440 + 242*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (11*n + 22)*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 + 242*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n + 40 + (11*n + 22)*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.