A038305 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(25 + 24\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 625 + 1200\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	625	1200	576															2401
1	15625	45000	43200	13824														117649
2	390625	1500000	2160000	1382400	331776													5764801
3	9765625	46875000	90000000	86400000	41472000	7962624												282475249
4	244140625	1406250000	3375000000	4320000000	3110400000	1194393600	191102976											13841287201
5	6103515625	41015625000	118125000000	189000000000	181440000000	104509440000	33443020800	4586471424										678223072849
6	152587890625	1171875000000	3937500000000	7560000000000	9072000000000	6967296000000	3344302080000	917294284800	110075314176									33232930569601
7	3814697265625	32958984375000	126562500000000	283500000000000	408240000000000	391910400000000	250822656000000	103195607040000	24766945689600	2641807540224								1628413597910449
8	95367431640625	915527343750000	3955078125000000	10125000000000000	17010000000000000	19595520000000000	15676416000000000	8599633920000000	3095868211200000	660451885056000	63403380965376							79792266297612001
9	2384185791015625	25177001953125000	120849609375000000	348046875000000000	668250000000000000	898128000000000000	862202880000000000	591224832000000000	283787919360000000	90812134195200000	17435929765478400	1521681143169024						3909821048582988049
10	59604644775390625	686645507812500000	3625488281250000000	11601562500000000000	25059375000000000000	38491200000000000000	43110144000000000000	35473489920000000000	21284093952000000000	9081213419520000000	2615389464821760000	456504342950707200	36520347436056576					191581231380566414401
11	1490116119384765625	18596649169921875000	107116699218750000000	377050781250000000000	904921875000000000000	1563705000000000000000	2001542400000000000000	1921480704000000000000	1383466106880000000000	737848590336000000000	283333858689024000000	74181955729489920000	11869112916718387200	876488338465357824				9387480337647754305649
12	37252902984619140625	500679016113281250000	3124237060546875000000	11997070312500000000000	31672265625000000000000	60810750000000000000000	87567480000000000000000	96074035200000000000000	80702189568000000000000	51649401323520000000000	24791712635289600000000	8654561501773824000000	2077094760425717760000	306770918462875238400	21035720123168587776			459986536544739960976801
13	931322574615478515625	13411045074462890625000	90122222900390625000000	374908447265625000000000	1079736328125000000000000	2280403125000000000000000	3648645000000000000000000	4503470400000000000000000	4323331584000000000000000	3228087582720000000000000	1859378447646720000000000	811365140791296000000000	259636845053214720000000	57519547211789107200000	7888395046188220416000	504857282956046106624		22539340290692258087863249
14	23283064365386962890625	357627868652343750000000	2574920654296875000000000	11535644531250000000000000	35991210937500000000000000	82923750000000000000000000	145945800000000000000000000	200154240000000000000000000	216166579200000000000000000	184462147584000000000000000	123958563176448000000000000	64909211263303680000000000	25963684505321472000000000	7669272961571880960000000	1577679009237644083200000	201942913182418442649600	12116574790945106558976	1104427674243920646305299201

Technical details

Node id: n-245f1aee5991880d · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-49793563bcf540fa

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:              625,              1200,                576;
  1:            15625,             45000,              43200,              13824;
  2:           390625,           1500000,            2160000,            1382400,             331776;
  3:          9765625,          46875000,           90000000,           86400000,           41472000,            7962624;
  4:        244140625,        1406250000,         3375000000,         4320000000,         3110400000,         1194393600,          191102976;
  5:       6103515625,       41015625000,       118125000000,       189000000000,       181440000000,       104509440000,        33443020800,         4586471424;
  6:     152587890625,     1171875000000,      3937500000000,      7560000000000,      9072000000000,      6967296000000,      3344302080000,       917294284800,       110075314176;
  7:    3814697265625,    32958984375000,    126562500000000,    283500000000000,    408240000000000,    391910400000000,    250822656000000,    103195607040000,     24766945689600,     2641807540224;
  8:   95367431640625,   915527343750000,   3955078125000000,  10125000000000000,  17010000000000000,  19595520000000000,  15676416000000000,   8599633920000000,   3095868211200000,   660451885056000,    63403380965376;
  9: 2384185791015625, 25177001953125000, 120849609375000000, 348046875000000000, 668250000000000000, 898128000000000000, 862202880000000000, 591224832000000000, 283787919360000000, 90812134195200000, 17435929765478400, 1521681143169024;

%F Formula

The row polynomials satisfy P(n,x) = (25+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 625 + 1200*x + 576*x^2.

%t Mathematica

P[0, x_] := 625 + 1200*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(25 + 24*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] = 625 + 1200*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (25 + 24*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool 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.