A013612 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(13 + 10\,x + 10\,n\right)\,P_{n}(x) = \left(138 + 520\,x + 200\,x^{2} + 60\,n + 200\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 276 + 1040\,x + 400\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	276	1040	400															1716
1	2376	16560	31200	8000														58136
2	18576	190080	662400	832000	160000													1863056
3	137376	1857600	9504000	22080000	20800000	3200000												57578976
4	979776	16485120	111456000	380160000	662400000	499200000	64000000											1734680896
5	6811776	137168640	1153958400	5201280000	13305600000	18547200000	11648000000	1280000000										51280018816
6	46469376	1089884160	10973491200	61544448000	208051200000	425779200000	494592000000	266240000000	25600000000									1493916692736
7	312408576	8364487680	98089574400	658409472000	2769500160000	7489843200000	12773376000000	12718080000000	5990400000000	512000000000								43018375302656
8	2076005376	62481715200	836448768000	6539304960000	32920473600000	110780006400000	249661440000000	364953600000000	317952000000000	133120000000000	10240000000000							1227067831448576
9	13665355776	456721182720	6872988672000	61339576320000	359661772800000	1448500838400000	4061933568000000	7846502400000000	10036224000000000	7772160000000000	2928640000000000	204800000000000						34727105530730496
10	89248075776	3279685386240	54806541926400	549839093760000	3680374579200000	17263765094400000	57940033536000000	139266293760000000	235395072000000000	267632640000000000	186531840000000000	63897600000000000	4096000000000000					976311633538748416
11	579024101376	23204499701760	426359100211200	4749900300288000	35739541094400000	191379478118400000	748096487424000000	2152058388480000000	4526154547200000000	6800302080000000000	6958448640000000000	4408934400000000000	1384448000000000000	81920000000000000				27292681605241102336
12	3735358488576	162126748385280	3248629958246400	39793516019712000	332493021020160000	2001414301286400000	8931042312192000000	29923859496960000000	75322043596800000000	140813697024000000000	190408458240000000000	177124147200000000000	102875136000000000000	29818880000000000000	1638400000000000000			759232779200343392256
13	23979434213376	1120607546572800	24319012257792000	324862995824640000	2984513701478400000	19949581261209600000	100070715064320000000	382758956236800000000	1122144731136000000000	2510734786560000000000	4224410910720000000000	5192957952000000000000	4428103680000000000000	2374041600000000000000	638976000000000000000	32768000000000000000		21030251753274871218176
14	153280304971776	7673418948280320	179297207451648000	2594027974164480000	25989039665971200000	191008876894617600000	1063977667264512000000	4574661260083200000000	15310358249472000000000	39898479329280000000000	80343513169920000000000	122891953766400000000000	138478878720000000000000	108999475200000000000000	54263808000000000000000	13631488000000000000000	655360000000000000000	580331732430861170180096

Technical details

Node id: n-2b1311b754401efd · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-3c9778e79ec3467a

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:         276,         1040,           400;
  1:        2376,        16560,         31200,           8000;
  2:       18576,       190080,        662400,         832000,          160000;
  3:      137376,      1857600,       9504000,       22080000,        20800000,          3200000;
  4:      979776,     16485120,     111456000,      380160000,       662400000,        499200000,         64000000;
  5:     6811776,    137168640,    1153958400,     5201280000,     13305600000,      18547200000,      11648000000,       1280000000;
  6:    46469376,   1089884160,   10973491200,    61544448000,    208051200000,     425779200000,     494592000000,     266240000000,       25600000000;
  7:   312408576,   8364487680,   98089574400,   658409472000,   2769500160000,    7489843200000,   12773376000000,   12718080000000,     5990400000000,     512000000000;
  8:  2076005376,  62481715200,  836448768000,  6539304960000,  32920473600000,  110780006400000,  249661440000000,  364953600000000,   317952000000000,  133120000000000,   10240000000000;
  9: 13665355776, 456721182720, 6872988672000, 61339576320000, 359661772800000, 1448500838400000, 4061933568000000, 7846502400000000, 10036224000000000, 7772160000000000, 2928640000000000, 204800000000000;

%F Formula

The row polynomials satisfy (10*n+13+10*x)*P(n,x) = (60*n+138+(200*n+520)*x+200*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 276 + 1040*x + 400*x^2.

%t Mathematica

P[0, x_] := 276 + 1040*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((60*n + 138 + (200*n + 520)*x + 200*x^2)*P[n - 1, x])/(10*n + 13 + 10*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 276 + 1040*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((60*n + 138 + (200*n + 520)*x + 200*x^2)*v[i - 1])/(10*n + 13 + 10*x)); 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+5x)^n.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.