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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-a5eb1303f4105072"]},{"id":183912,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-a5eb1303f4105072"]},{"id":183913,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"77c3e625ff8a7e09b5b43fd723ef412666afec5b2d5cc2c43724b0cdbf536a77","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-14be0ce6edbde2a1","role":"sequence","name":"Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     2,       1;\n  2:     9,      10,        1;\n  3:    32,      93,       34,         1;\n  4:   103,     754,      632,       106,         1;\n  5:   318,    5129,     9624,      3674,       322,         1;\n  6:   965,   31138,   118095,     94684,     19907,       970,        1;\n  7:  2908,  176253,  1229514,   1904735,    814312,    104115,     2914,       1;\n  8:  8739,  954906, 11423702,  31353078,  25176748,   6488710,   534250,    8746,     1;\n  9: 26234, 5030841, 98126944, 444857368, 619473732, 294494902, 49391008, 2712168, 26242, 1;\n","recurrence_sentence":"The row polynomials satisfy 16*P(n,x) = (48*n^2-128*n+52+(24*n^2-64*n-70)*x-112*x^2-(48*n^2-176*n+108)*x^3)*P(n-4,x) + ((24*n^2-64*n+26)*x-48*x^2-(48*n^2-176*n+108)*x^3)*d/dx P(n-4,x) + (48*n^3-284*n^2+468*n-169+(24*n^3-76*n^2-134*n+475)*x-(24*n^2+88*n-418)*x^2-(48*n^3-296*n^2+548*n-142)*x^3)*P(n-3,x) + (60*n^2-160*n+65+(24*n^3-196*n^2+378*n-263)*x-(48*n^2-104*n-184)*x^2-(48*n^3-344*n^2+724*n-426)*x^3+(48*n^2-176*n+108)*x^4)*d/dx P(n-3,x) + ((24*n^2-64*n+26)*x-(24*n^2-64*n+74)*x^2-(48*n^2-176*n+60)*x^3+(48*n^2-176*n+108)*x^4)*d^2/dx^2 P(n-3,x) - (12*n^2+40*n-23+(96*n^2-464*n+200)*x+(40*n^2-200*n+298)*x^2+(128*n-192)*x^3)*P(n-2,x) - (12*n^2-32*n+13+(48*n-208)*x-(24*n^2-40*n-226)*x^2+32*x^3-128*x^4)*d/dx P(n-2,x) + (64*x^2-128*x^3+64*x^4)*d^2/dx^2 P(n-2,x) + (24*n+36+(64*n-208)*x+64*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 9 + 10*x + x^2, P(3,x) = 32 + 93*x + 34*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 9 + 10*x + x^2;\nP[3, x_] := 32 + 93*x + 34*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((48*n^2 - 128*n + 52 + (24*n^2 - 64*n - 70)*x - 112*x^2 - (48*n^2 - 176*n + 108)*x^3)*P[n - 4, x] + ((24*n^2 - 64*n + 26)*x - 48*x^2 - (48*n^2 - 176*n + 108)*x^3)*D[P[n - 4, x], x] + (48*n^3 - 284*n^2 + 468*n - 169 + (24*n^3 - 76*n^2 - 134*n + 475)*x - (24*n^2 + 88*n - 418)*x^2 - (48*n^3 - 296*n^2 + 548*n - 142)*x^3)*P[n - 3, x] + (60*n^2 - 160*n + 65 + (24*n^3 - 196*n^2 + 378*n - 263)*x - (48*n^2 - 104*n - 184)*x^2 - (48*n^3 - 344*n^2 + 724*n - 426)*x^3 + (48*n^2 - 176*n + 108)*x^4)*D[P[n - 3, x], x] + ((24*n^2 - 64*n + 26)*x - (24*n^2 - 64*n + 74)*x^2 - (48*n^2 - 176*n + 60)*x^3 + (48*n^2 - 176*n + 108)*x^4)*D[P[n - 3, x], {x, 2}] - (12*n^2 + 40*n - 23 + (96*n^2 - 464*n + 200)*x + (40*n^2 - 200*n + 298)*x^2 + (128*n - 192)*x^3)*P[n - 2, x] - (12*n^2 - 32*n + 13 + (48*n - 208)*x - (24*n^2 - 40*n - 226)*x^2 + 32*x^3 - 128*x^4)*D[P[n - 2, x], x] + (64*x^2 - 128*x^3 + 64*x^4)*D[P[n - 2, x], {x, 2}] + (24*n + 36 + (64*n - 208)*x + 64*x^2)*P[n - 1, x])/(16)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 9 + 10*x + x^2; v[4] = 32 + 93*x + 34*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((48*n^2 - 128*n + 52 + (24*n^2 - 64*n - 70)*x - 112*x^2 - (48*n^2 - 176*n + 108)*x^3)*v[i - 4] + ((24*n^2 - 64*n + 26)*x - 48*x^2 - (48*n^2 - 176*n + 108)*x^3)*deriv(v[i - 4], x) + (48*n^3 - 284*n^2 + 468*n - 169 + (24*n^3 - 76*n^2 - 134*n + 475)*x - (24*n^2 + 88*n - 418)*x^2 - (48*n^3 - 296*n^2 + 548*n - 142)*x^3)*v[i - 3] + (60*n^2 - 160*n + 65 + (24*n^3 - 196*n^2 + 378*n - 263)*x - (48*n^2 - 104*n - 184)*x^2 - (48*n^3 - 344*n^2 + 724*n - 426)*x^3 + (48*n^2 - 176*n + 108)*x^4)*deriv(v[i - 3], x) + ((24*n^2 - 64*n + 26)*x - (24*n^2 - 64*n + 74)*x^2 - (48*n^2 - 176*n + 60)*x^3 + (48*n^2 - 176*n + 108)*x^4)*deriv(deriv(v[i - 3], x), x) - (12*n^2 + 40*n - 23 + (96*n^2 - 464*n + 200)*x + (40*n^2 - 200*n + 298)*x^2 + (128*n - 192)*x^3)*v[i - 2] - (12*n^2 - 32*n + 13 + (48*n - 208)*x - (24*n^2 - 40*n - 226)*x^2 + 32*x^3 - 128*x^4)*deriv(v[i - 2], x) + (64*x^2 - 128*x^3 + 64*x^4)*deriv(deriv(v[i - 2], x), x) + (24*n + 36 + (64*n - 208)*x + 64*x^2)*v[i - 1])/(16)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-14be0ce6edbde2a1","steps":["D+1"],"edges":["e-a5eb1303f4105072"],"text":"A144439 \u2192 D+1"},"headline":"A144439 \u2192 D+1","chain":null}