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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-3252facb45670343"]},{"id":174968,"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-3252facb45670343"]},{"id":174969,"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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n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"b623fea8c4dd8734379172f89a12227f008bef76373c963e0b8fb2003aebb102","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a88a492d75692c65","role":"sequence","name":"Triangle for first differences of Catalan numbers.","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:   0;\n  1:   1;\n  2:   5,    2;\n  3:  14,   18,     3;\n  4:  30,   80,    42,     4;\n  5:  55,  250,   270,    80,     5;\n  6:  91,  630,  1155,   700,   135,     6;\n  7: 140, 1372,  3822,  3920,  1540,   210,    7;\n  8: 204, 2688, 10584, 16464, 10920,  3024,  308,   8;\n  9: 285, 4860, 25704, 56448, 56700, 26460, 5460, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+22*n^2+34*n+12)*P(n,x) = (60*n^3+148*n^2-1226*n-1304-(84*n^3+396*n^2-2226*n-2808)*x-(12*n^3-348*n^2+774*n+1704)*x^2+(36*n^3-100*n^2-226*n+200)*x^3)*P(n-5,x) - (16*n^3+56*n^2-375*n-420-(32*n^3+208*n^2-1016*n-1312)*x+(288*n^2-798*n-1416)*x^2+(32*n^3-176*n^2+48*n+576)*x^3-(16*n^3-40*n^2-109*n+52)*x^4)*d/dx P(n-5,x) - (52*n^3-514*n^2+704*n+1516-(208*n^3+4*n^2-704*n+114)*x-(12*n^3-294*n^2+168*n+600)*x^2+(40*n^3-32*n^2-168*n+118)*x^3)*P(n-4,x) + (64*n^3+176*n^2-504*n-582-(32*n^3+160*n^2-164*n-391)*x-(16*n^3-56*n^2-176*n-9)*x^2-(80*n^2-100*n-217)*x^3-(16*n^3-8*n^2-64*n+35)*x^4)*d/dx P(n-4,x) - (72*n^3+712*n^2+338*n-336+(152*n^3-268*n^2-430*n+330)*x-(12*n^3-24*n^2-102*n+60)*x^2-(4*n^3+4*n^2-14*n+6)*x^3)*P(n-3,x) - (96*n^3+192*n^2+18*n-6+(32*n^3-80*n^2-24*n+150)*x-(32*n^3+272*n^2+71*n-123)*x^2-(32*n^3+32*n^2-58*n+24)*x^3+(3*n-3)*x^4)*d/dx P(n-3,x) + (56*n^3+36*n^2-22*n+8+(16*n^3+76*n^2+100*n+24)*x-(12*n^3+30*n^2+12*n)*x^2)*P(n-2,x) + (64*n^3+80*n^2+48*n+54+(32*n^3-128*n^2-316*n-161)*x-(16*n^3+40*n^2-14*n-24)*x^2+(6*n+3)*x^3)*d/dx P(n-2,x) + (12*n^3+36*n^2+20*n-8+(12*n^3+48*n^2+60*n+24)*x)*P(n-1,x) - (16*n^3+8*n^2-99*n-102+(8*n+16)*x+(3*n+6)*x^2)*d/dx P(n-1,x) for n >= 5, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x, P(3,x) = 14 + 18*x + 3*x^2, P(4,x) = 30 + 80*x + 42*x^2 + 4*x^3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 5 + 2*x;\nP[3, x_] := 14 + 18*x + 3*x^2;\nP[4, x_] := 30 + 80*x + 42*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((60*n^3 + 148*n^2 - 1226*n - 1304 - (84*n^3 + 396*n^2 - 2226*n - 2808)*x - (12*n^3 - 348*n^2 + 774*n + 1704)*x^2 + (36*n^3 - 100*n^2 - 226*n + 200)*x^3)*P[n - 5, x] - (16*n^3 + 56*n^2 - 375*n - 420 - (32*n^3 + 208*n^2 - 1016*n - 1312)*x + (288*n^2 - 798*n - 1416)*x^2 + (32*n^3 - 176*n^2 + 48*n + 576)*x^3 - (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*D[P[n - 5, x], x] - (52*n^3 - 514*n^2 + 704*n + 1516 - (208*n^3 + 4*n^2 - 704*n + 114)*x - (12*n^3 - 294*n^2 + 168*n + 600)*x^2 + (40*n^3 - 32*n^2 - 168*n + 118)*x^3)*P[n - 4, x] + (64*n^3 + 176*n^2 - 504*n - 582 - (32*n^3 + 160*n^2 - 164*n - 391)*x - (16*n^3 - 56*n^2 - 176*n - 9)*x^2 - (80*n^2 - 100*n - 217)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*D[P[n - 4, x], x] - (72*n^3 + 712*n^2 + 338*n - 336 + (152*n^3 - 268*n^2 - 430*n + 330)*x - (12*n^3 - 24*n^2 - 102*n + 60)*x^2 - (4*n^3 + 4*n^2 - 14*n + 6)*x^3)*P[n - 3, x] - (96*n^3 + 192*n^2 + 18*n - 6 + (32*n^3 - 80*n^2 - 24*n + 150)*x - (32*n^3 + 272*n^2 + 71*n - 123)*x^2 - (32*n^3 + 32*n^2 - 58*n + 24)*x^3 + (3*n - 3)*x^4)*D[P[n - 3, x], x] + (56*n^3 + 36*n^2 - 22*n + 8 + (16*n^3 + 76*n^2 + 100*n + 24)*x - (12*n^3 + 30*n^2 + 12*n)*x^2)*P[n - 2, x] + (64*n^3 + 80*n^2 + 48*n + 54 + (32*n^3 - 128*n^2 - 316*n - 161)*x - (16*n^3 + 40*n^2 - 14*n - 24)*x^2 + (6*n + 3)*x^3)*D[P[n - 2, x], x] + (12*n^3 + 36*n^2 + 20*n - 8 + (12*n^3 + 48*n^2 + 60*n + 24)*x)*P[n - 1, x] - (16*n^3 + 8*n^2 - 99*n - 102 + (8*n + 16)*x + (3*n + 6)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 22*n^2 + 34*n + 12)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 5 + 2*x; v[4] = 14 + 18*x + 3*x^2; v[5] = 30 + 80*x + 42*x^2 + 4*x^3; for(n = 5, N, my(i = n + 1); v[i] = ((60*n^3 + 148*n^2 - 1226*n - 1304 - (84*n^3 + 396*n^2 - 2226*n - 2808)*x - (12*n^3 - 348*n^2 + 774*n + 1704)*x^2 + (36*n^3 - 100*n^2 - 226*n + 200)*x^3)*v[i - 5] - (16*n^3 + 56*n^2 - 375*n - 420 - (32*n^3 + 208*n^2 - 1016*n - 1312)*x + (288*n^2 - 798*n - 1416)*x^2 + (32*n^3 - 176*n^2 + 48*n + 576)*x^3 - (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*deriv(v[i - 5], x) - (52*n^3 - 514*n^2 + 704*n + 1516 - (208*n^3 + 4*n^2 - 704*n + 114)*x - (12*n^3 - 294*n^2 + 168*n + 600)*x^2 + (40*n^3 - 32*n^2 - 168*n + 118)*x^3)*v[i - 4] + (64*n^3 + 176*n^2 - 504*n - 582 - (32*n^3 + 160*n^2 - 164*n - 391)*x - (16*n^3 - 56*n^2 - 176*n - 9)*x^2 - (80*n^2 - 100*n - 217)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*deriv(v[i - 4], x) - (72*n^3 + 712*n^2 + 338*n - 336 + (152*n^3 - 268*n^2 - 430*n + 330)*x - (12*n^3 - 24*n^2 - 102*n + 60)*x^2 - (4*n^3 + 4*n^2 - 14*n + 6)*x^3)*v[i - 3] - (96*n^3 + 192*n^2 + 18*n - 6 + (32*n^3 - 80*n^2 - 24*n + 150)*x - (32*n^3 + 272*n^2 + 71*n - 123)*x^2 - (32*n^3 + 32*n^2 - 58*n + 24)*x^3 + (3*n - 3)*x^4)*deriv(v[i - 3], x) + (56*n^3 + 36*n^2 - 22*n + 8 + (16*n^3 + 76*n^2 + 100*n + 24)*x - (12*n^3 + 30*n^2 + 12*n)*x^2)*v[i - 2] + (64*n^3 + 80*n^2 + 48*n + 54 + (32*n^3 - 128*n^2 - 316*n - 161)*x - (16*n^3 + 40*n^2 - 14*n - 24)*x^2 + (6*n + 3)*x^3)*deriv(v[i - 2], x) + (12*n^3 + 36*n^2 + 20*n - 8 + (12*n^3 + 48*n^2 + 60*n + 24)*x)*v[i - 1] - (16*n^3 + 8*n^2 - 99*n - 102 + (8*n + 16)*x + (3*n + 6)*x^2)*deriv(v[i - 1], x))/(4*n^3 + 22*n^2 + 34*n + 12)); 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-a88a492d75692c65","steps":["D"],"edges":["e-3252facb45670343"],"text":"A119308 \u2192 D"},"headline":"A119308 \u2192 D","chain":null}