A395972 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 + x + x^{2} + 4\,n x + 4\,n x^{2} + 3\,n^{2} x + 6\,n^{2} x^{2} + 4\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(2 + 10\,x + 10\,x^{2} + 8\,x^{3} + 24\,n x + 56\,n x^{2} + 48\,n x^{3} + 6\,n^{2} x + 76\,n^{2} x^{2} + 94\,n^{2} x^{3} + 28\,n^{3} x^{2} + 80\,n^{3} x^{3} + 2\,n^{4} x^{2} + 30\,n^{4} x^{3} + 4\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 12\,x^{3} - 16\,x^{4} - 20\,n x^{3} - 44\,n x^{4} - 6\,n^{2} x^{3} - 42\,n^{2} x^{4} - 16\,n^{3} x^{4} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 4 + 64\,x + 72\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 4 | 64 | 72 | 140 | |||||||
| 2 | 8 | 288 | 1296 | 768 | 2360 | ||||||
| 3 | 16 | 1024 | 10368 | 24576 | 9600 | 45584 | |||||
| 4 | 32 | 3200 | 57600 | 307200 | 480000 | 138240 | 986272 | ||||
| 5 | 64 | 9216 | 259200 | 2457600 | 8640000 | 9953280 | 2257920 | 23577280 | |||
| 6 | 128 | 25088 | 1016064 | 15052800 | 94080000 | 243855360 | 221276160 | 41287680 | 616593280 | ||
| 7 | 256 | 65536 | 3612672 | 77070336 | 752640000 | 3468165120 | 7080837120 | 5284823040 | 836075520 | 17503289600 | |
| 8 | 512 | 165888 | 11943936 | 346816512 | 4877107200 | 35115171840 | 127455068160 | 214035333120 | 135444234240 | 18579456000 | 535865297408 |
Compute more
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Technical details
- Node id
n-b947e3321c316ff1· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-53219bd9184fb09a
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 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): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 4, 64, 72; 2: 8, 288, 1296, 768; 3: 16, 1024, 10368, 24576, 9600; 4: 32, 3200, 57600, 307200, 480000, 138240; 5: 64, 9216, 259200, 2457600, 8640000, 9953280, 2257920; 6: 128, 25088, 1016064, 15052800, 94080000, 243855360, 221276160, 41287680; 7: 256, 65536, 3612672, 77070336, 752640000, 3468165120, 7080837120, 5284823040, 836075520; 8: 512, 165888, 11943936, 346816512, 4877107200, 35115171840, 127455068160, 214035333120, 135444234240, 18579456000; 9: 1024, 409600, 37324800, 1415577600, 27095040000, 280921374720, 1593188352000, 4756340736000, 6772211712000, 3715891200000, 449622835200; 10: 2048, 991232, 111513600, 5352652800, 133816320000, 1888415907840, 15422063247360, 71939653632000, 182097248256000, 224811417600000, 108808726118400, 11771943321600;
%F FormulaThe row polynomials satisfy (1+(3*n^2+4*n+1)*x+(n^4+4*n^3+6*n^2+4*n+1)*x^2)*P(n,x) = (2+(6*n^2+24*n+10)*x+(2*n^4+28*n^3+76*n^2+56*n+10)*x^2+(4*n^5+30*n^4+80*n^3+94*n^2+48*n+8)*x^3)*P(n-1,x) - (2*x^2+(6*n^2+20*n+12)*x^3+(2*n^4+16*n^3+42*n^2+44*n+16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 64*x + 72*x^2.
%t MathematicaP[1, x_] := 4 + 64*x + 72*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n^2 + 24*n + 10)*x + (2*n^4 + 28*n^3 + 76*n^2 + 56*n + 10)*x^2 + (4*n^5 + 30*n^4 + 80*n^3 + 94*n^2 + 48*n + 8)*x^3)*P[n - 1, x] - (2*x^2 + (6*n^2 + 20*n + 12)*x^3 + (2*n^4 + 16*n^3 + 42*n^2 + 44*n + 16)*x^4)*D[P[n - 1, x], x])/(1 + (3*n^2 + 4*n + 1)*x + (n^4 + 4*n^3 + 6*n^2 + 4*n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 4 + 64*x + 72*x^2; for(n = 2, N, my(i = n); v[i] = ((2 + (6*n^2 + 24*n + 10)*x + (2*n^4 + 28*n^3 + 76*n^2 + 56*n + 10)*x^2 + (4*n^5 + 30*n^4 + 80*n^3 + 94*n^2 + 48*n + 8)*x^3)*v[i - 1] - (2*x^2 + (6*n^2 + 20*n + 12)*x^3 + (2*n^4 + 16*n^3 + 42*n^2 + 44*n + 16)*x^4)*deriv(v[i - 1], x))/(1 + (3*n^2 + 4*n + 1)*x + (n^4 + 4*n^3 + 6*n^2 + 4*n + 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, 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 ≥ 2
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).?
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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
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from
Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).?
Related ?
Same fingerprint ?
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.