A144441 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(32\right)\,P_{n}(x) = \left(824 - 356\,x - 896\,x^{2} - 2736\,x^{3} - 1216\,n - 608\,n x + 3200\,n x^{3} + 384\,n^{2} + 192\,n^{2} x - 768\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(412\,x - 384\,x^{2} - 2736\,x^{3} - 608\,n x + 3200\,n x^{3} + 192\,n^{2} x - 768\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-2987 + 4331\,x + 3444\,x^{2} + 6500\,x^{3} + 5232\,n - 668\,n x - 608\,n x^{2} - 11536\,n x^{3} - 2608\,n^{2} - 752\,n^{2} x - 192\,n^{2} x^{2} + 5312\,n^{2} x^{3} + 384\,n^{3} + 192\,n^{3} x - 768\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(927 - 3233\,x + 1112\,x^{2} + 10644\,x^{3} + 2736\,x^{4} - 1368\,n + 3908\,n x + 1024\,n x^{2} - 14736\,n x^{3} - 3200\,n x^{4} + 432\,n^{2} - 1712\,n^{2} x - 384\,n^{2} x^{2} + 6080\,n^{2} x^{3} + 768\,n^{2} x^{4} + 192\,n^{3} x - 768\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(412\,x - 796\,x^{2} - 2352\,x^{3} + 2736\,x^{4} - 608\,n x + 608\,n x^{2} + 3200\,n x^{3} - 3200\,n x^{4} + 192\,n^{2} x - 192\,n^{2} x^{2} - 768\,n^{2} x^{3} + 768\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(417 - 4080\,x - 2484\,x^{2} + 1792\,x^{3} - 328\,n + 4352\,n x + 1760\,n x^{2} - 1024\,n x^{3} - 48\,n^{2} - 768\,n^{2} x - 320\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-103 + 1664\,x - 1652\,x^{2} - 256\,x^{3} + 1024\,x^{4} + 152\,n - 384\,n x - 416\,n x^{2} - 48\,n^{2} + 192\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(512\,x^{2} - 1024\,x^{3} + 512\,x^{4}\right)\,P_{n-2}''(x) + \left(56 - 896\,x + 256\,x^{2} + 96\,n + 256\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 15 + 16\,x + x^{2}\)
\(P_{3}(x) = 84 + 249\,x + 86\,x^{2} + x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 15 | 16 | 1 | 32 | |||||
| 3 | 84 | 249 | 86 | 1 | 420 | ||||
| 4 | 433 | 3532 | 2846 | 436 | 1 | 7248 | |||
| 5 | 2182 | 41977 | 79592 | 28622 | 2186 | 1 | 154560 | ||
| 6 | 10931 | 446704 | 1788531 | 1395096 | 272537 | 10936 | 1 | 3924736 | |
| 7 | 54680 | 4453153 | 33968622 | 52948975 | 21585980 | 2527311 | 54686 | 1 | 115593408 |
Compute more
?
Technical details
- Node id
n-4d4d4fc04fd9c447· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-7cb3cefa1e2361e4
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
- 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 ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 15, 16, 1; 3: 84, 249, 86, 1; 4: 433, 3532, 2846, 436, 1; 5: 2182, 41977, 79592, 28622, 2186, 1; 6: 10931, 446704, 1788531, 1395096, 272537, 10936, 1; 7: 54680, 4453153, 33968622, 52948975, 21585980, 2527311, 54686, 1; 8: 273429, 42681084, 575372408, 1633311748, 1285761522, 311945828, 23117824, 273436, 1; 9: 1367178, 399332337, 9019852064, 43252457884, 60468252084, 27719780982, 4333167888, 209919996, 1367186, 1;
%F FormulaThe row polynomials satisfy 32*P(n,x) = (384*n^2-1216*n+824+(192*n^2-608*n-356)*x-896*x^2-(768*n^2-3200*n+2736)*x^3)*P(n-4,x) + ((192*n^2-608*n+412)*x-384*x^2-(768*n^2-3200*n+2736)*x^3)*d/dx P(n-4,x) + (384*n^3-2608*n^2+5232*n-2987+(192*n^3-752*n^2-668*n+4331)*x-(192*n^2+608*n-3444)*x^2-(768*n^3-5312*n^2+11536*n-6500)*x^3)*P(n-3,x) + (432*n^2-1368*n+927+(192*n^3-1712*n^2+3908*n-3233)*x-(384*n^2-1024*n-1112)*x^2-(768*n^3-6080*n^2+14736*n-10644)*x^3+(768*n^2-3200*n+2736)*x^4)*d/dx P(n-3,x) + ((192*n^2-608*n+412)*x-(192*n^2-608*n+796)*x^2-(768*n^2-3200*n+2352)*x^3+(768*n^2-3200*n+2736)*x^4)*d^2/dx^2 P(n-3,x) - (48*n^2+328*n-417+(768*n^2-4352*n+4080)*x+(320*n^2-1760*n+2484)*x^2+(1024*n-1792)*x^3)*P(n-2,x) - (48*n^2-152*n+103+(384*n-1664)*x-(192*n^2-416*n-1652)*x^2+256*x^3-1024*x^4)*d/dx P(n-2,x) + (512*x^2-1024*x^3+512*x^4)*d^2/dx^2 P(n-2,x) + (96*n+56+(256*n-896)*x+256*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 15 + 16*x + x^2, P(3,x) = 84 + 249*x + 86*x^2 + x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 15 + 16*x + x^2;
P[3, x_] := 84 + 249*x + 86*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((384*n^2 - 1216*n + 824 + (192*n^2 - 608*n - 356)*x - 896*x^2 - (768*n^2 - 3200*n + 2736)*x^3)*P[n - 4, x] + ((192*n^2 - 608*n + 412)*x - 384*x^2 - (768*n^2 - 3200*n + 2736)*x^3)*D[P[n - 4, x], x] + (384*n^3 - 2608*n^2 + 5232*n - 2987 + (192*n^3 - 752*n^2 - 668*n + 4331)*x - (192*n^2 + 608*n - 3444)*x^2 - (768*n^3 - 5312*n^2 + 11536*n - 6500)*x^3)*P[n - 3, x] + (432*n^2 - 1368*n + 927 + (192*n^3 - 1712*n^2 + 3908*n - 3233)*x - (384*n^2 - 1024*n - 1112)*x^2 - (768*n^3 - 6080*n^2 + 14736*n - 10644)*x^3 + (768*n^2 - 3200*n + 2736)*x^4)*D[P[n - 3, x], x] + ((192*n^2 - 608*n + 412)*x - (192*n^2 - 608*n + 796)*x^2 - (768*n^2 - 3200*n + 2352)*x^3 + (768*n^2 - 3200*n + 2736)*x^4)*D[P[n - 3, x], {x, 2}] - (48*n^2 + 328*n - 417 + (768*n^2 - 4352*n + 4080)*x + (320*n^2 - 1760*n + 2484)*x^2 + (1024*n - 1792)*x^3)*P[n - 2, x] - (48*n^2 - 152*n + 103 + (384*n - 1664)*x - (192*n^2 - 416*n - 1652)*x^2 + 256*x^3 - 1024*x^4)*D[P[n - 2, x], x] + (512*x^2 - 1024*x^3 + 512*x^4)*D[P[n - 2, x], {x, 2}] + (96*n + 56 + (256*n - 896)*x + 256*x^2)*P[n - 1, x])/(32)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 15 + 16*x + x^2; v[4] = 84 + 249*x + 86*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((384*n^2 - 1216*n + 824 + (192*n^2 - 608*n - 356)*x - 896*x^2 - (768*n^2 - 3200*n + 2736)*x^3)*v[i - 4] + ((192*n^2 - 608*n + 412)*x - 384*x^2 - (768*n^2 - 3200*n + 2736)*x^3)*deriv(v[i - 4], x) + (384*n^3 - 2608*n^2 + 5232*n - 2987 + (192*n^3 - 752*n^2 - 668*n + 4331)*x - (192*n^2 + 608*n - 3444)*x^2 - (768*n^3 - 5312*n^2 + 11536*n - 6500)*x^3)*v[i - 3] + (432*n^2 - 1368*n + 927 + (192*n^3 - 1712*n^2 + 3908*n - 3233)*x - (384*n^2 - 1024*n - 1112)*x^2 - (768*n^3 - 6080*n^2 + 14736*n - 10644)*x^3 + (768*n^2 - 3200*n + 2736)*x^4)*deriv(v[i - 3], x) + ((192*n^2 - 608*n + 412)*x - (192*n^2 - 608*n + 796)*x^2 - (768*n^2 - 3200*n + 2352)*x^3 + (768*n^2 - 3200*n + 2736)*x^4)*deriv(deriv(v[i - 3], x), x) - (48*n^2 + 328*n - 417 + (768*n^2 - 4352*n + 4080)*x + (320*n^2 - 1760*n + 2484)*x^2 + (1024*n - 1792)*x^3)*v[i - 2] - (48*n^2 - 152*n + 103 + (384*n - 1664)*x - (192*n^2 - 416*n - 1652)*x^2 + 256*x^3 - 1024*x^4)*deriv(v[i - 2], x) + (512*x^2 - 1024*x^3 + 512*x^4)*deriv(deriv(v[i - 2], x), x) + (96*n + 56 + (256*n - 896)*x + 256*x^2)*v[i - 1])/(32)); 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]
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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
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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
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n,…?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n,…?
References
- [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.