A038321 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 + 210\,x + 196\,x^{2} + 126\,n x + 196\,n x^{2} + 49\,n^{2} x^{2}\right)\,P_{n}(x) = \left(648 + 6804\,x + 14994\,x^{2} + 9261\,x^{3} + 2268\,n x + 7938\,n x^{2} + 6174\,n x^{3} + 882\,n^{2} x^{2} + 1029\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 324 + 3024\,x + 3969\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	324	3024	3969															7317
1	5832	81648	214326	148176														449982
2	104976	1959552	7715736	10668672	4862025													25310961
3	1889568	44089920	231472080	480090240	437582250	147027636												1342151694
4	34012224	952342272	6249746160	17283248640	23629441500	15878984688	4202539929											68230315413
5	612220032	19999187712	157493603232	544422332160	992436543000	1000376035344	529520031054	115269666624										3360129619158
6	11019960576	411411861504	3779846477568	15679363166208	35727715548000	48018049696512	38125442235888	16598831993856	3063651608241									161415332548353
7	198359290368	8331090195456	87476447052288	423342805487616	1157577983755200	1944731012708736	2058773880737952	1344505391502336	496311560535042	79428004658100								7600676535923094
8	3570467226624	166621803909120	1968220058676480	10885957855395840	34727339512656000	70010316457514496	92644824633207840	80670323490140160	44668040448153780	14297040838458000	2018265598362321							352060521163700661
9	64268410079232	3299111717400576	43300841290882560	269427456921047040	982287603357984000	2310340443097978368	3668735055475030464	3993181012761937920	2948090669578149480	1415407043007342000	399616588475739558	50439960078079824						16084190054171651022
10	1156831381426176	64782557359865856	935298171883063296	6466258966105128960	26521765290665568000	71290505101309046784	132074461997101096704	172505419751315718144	159196896157220071920	101909307096528624000	43158591555379872264	10895031376865241984	1243134849424342329					726262609702539066417
11	20822964865671168	1263259868517384192	19896342929148801024	151310459806860017664	689565897557304768000	2085247274213289618432	4415060586760236661248	6727711370301313007616	7450414740157899365856	5961694465146924504000	3366370141319630036592	1274718671093233312128	290893554765296104986	30276586036867414356				32464444172921386667262
12	374813367582081024	24487806682029293568	417823201512124821504	3466385079211702222848	17377060618444080153600	58386923677972109316096	139074408482947454829312	242197609330847268274176	312917419086631773365952	300469401043404995001600	212081318903136692305296	107076368371831598218752	36652587900427309228236	7629699681290588417712	729881984817339453225			1438501749982524646982901
13	6746640616477458432	472264843153422090240	8677866492944130908160	77993664282263300014080	426527851543627421952000	1576446939305246951534592	4172232254488423644879360	8174169314916095304253440	12069671879055796972686720	13521123046953224775072000	11452391220769381384485984	7227654865098632879765760	3298732911038457830541240	1030009456974229436391120	197068135900681652370750	17439313557235630669056		63250617731860011215072934
14	121439531096594251776	9067484988545704132608	178516110711993550110720	1727859639483986954158080	10236668437047058126848000	41274247138173738367451136	120160288929266600972525568	261573418077315049736110080	434508187646008691016721920	556297633931789819317248000	549714778596930306455327232	416312920229681253874507776	237508769594768963798969280	98880907869526025893547520	28377811569698157941388000	5022522304483861632688128	413434351479933759884769	2762197153350885083695870593

Technical details

Node id: n-88426f8cec0f73a5 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d8ab6700f3e17b80

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            324,             3024,              3969;
  1:           5832,            81648,            214326,             148176;
  2:         104976,          1959552,           7715736,           10668672,            4862025;
  3:        1889568,         44089920,         231472080,          480090240,          437582250,           147027636;
  4:       34012224,        952342272,        6249746160,        17283248640,        23629441500,         15878984688,          4202539929;
  5:      612220032,      19999187712,      157493603232,       544422332160,       992436543000,       1000376035344,        529520031054,        115269666624;
  6:    11019960576,     411411861504,     3779846477568,     15679363166208,     35727715548000,      48018049696512,      38125442235888,      16598831993856,       3063651608241;
  7:   198359290368,    8331090195456,    87476447052288,    423342805487616,   1157577983755200,    1944731012708736,    2058773880737952,    1344505391502336,     496311560535042,      79428004658100;
  8:  3570467226624,  166621803909120,  1968220058676480,  10885957855395840,  34727339512656000,   70010316457514496,   92644824633207840,   80670323490140160,   44668040448153780,   14297040838458000,   2018265598362321;
  9: 64268410079232, 3299111717400576, 43300841290882560, 269427456921047040, 982287603357984000, 2310340443097978368, 3668735055475030464, 3993181012761937920, 2948090669578149480, 1415407043007342000, 399616588475739558, 50439960078079824;

%F Formula

The row polynomials satisfy (36+(126*n+210)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (648+(2268*n+6804)*x+(882*n^2+7938*n+14994)*x^2+(1029*n^2+6174*n+9261)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 3024*x + 3969*x^2.

%t Mathematica

P[0, x_] := 324 + 3024*x + 3969*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((648 + (2268*n + 6804)*x + (882*n^2 + 7938*n + 14994)*x^2 + (1029*n^2 + 6174*n + 9261)*x^3)*P[n - 1, x])/(36 + (126*n + 210)*x + (49*n^2 + 196*n + 196)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 3024*x + 3969*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((648 + (2268*n + 6804)*x + (882*n^2 + 7938*n + 14994)*x^2 + (1029*n^2 + 6174*n + 9261)*x^3)*v[i - 1])/(36 + (126*n + 210)*x + (49*n^2 + 196*n + 196)*x^2)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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
    
    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.