{"id":"n-9feefe9ab18e6aba","revision":0,"first_index":0,"fingerprint":"f6-c99ae9576f2717ab","name":"derivative of basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["9"],["18"],["81","162"],["198","486"],["612","2430","2187"],["1566","7452","8748"],["4275","25758","48114","26244"],["10926","76302","174960","131220"],["28323","230526","667035","787320","295245"],["71532","656100","2221992","3280500","1771470"],["180648","1857492","7335198","13646880","11514555","3188646"],["450684","5113368","22858524","50152284","53734590","22320522"],["1120509","13937508","69940260","179115300","241805655","156243654","33480783"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","9","18","243","684","5229","17766","104391","393408","2008449","8001594","37723419","154629972","695643669"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["9"]],"recurrence":{"denominator":{"coeffs":[["-6"],["3"]]},"terms":[{"coeff":{"coeffs":[["-57","-162"],["21","54"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-13","-75","-108"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175540,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 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-9995913d3601081a"]},{"id":175541,"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-9995913d3601081a"]},{"id":175542,"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 real-rootedness)","region_exact":null,"edges":["e-9995913d3601081a"]}],"region_exact":{},"incoming_edges":[{"id":"e-9995913d3601081a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-9feefe9ab18e6aba","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a 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":"f3dccfee727e4c18162c73a976e362d0c7403e4f49ca9a467a97c16ce23b6264","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-8d5fed5e270db201","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, 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:     0;\n  1:     0;\n  2:     9;\n  3:    18;\n  4:    81,   162;\n  5:   198,   486;\n  6:   612,  2430,   2187;\n  7:  1566,  7452,   8748;\n  8:  4275, 25758,  48114,  26244;\n  9: 10926, 76302, 174960, 131220;\n","recurrence_sentence":"The row polynomials satisfy (3*n-6)*P(n,x) = (21*n-57+(54*n-162)*x)*P(n-1,x) - (13+75*x+108*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 9.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 9;\nP[n_, x_] := P[n, x] = Expand[Together[((21*n - 57 + (54*n - 162)*x)*P[n - 1, x] - (13 + 75*x + 108*x^2)*D[P[n - 1, x], x])/(3*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 9; for(n = 3, N, my(i = n + 1); v[i] = ((21*n - 57 + (54*n - 162)*x)*v[i - 1] - (13 + 75*x + 108*x^2)*deriv(v[i - 1], x))/(3*n - 6)); 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-e189038e95d51606","steps":["basis A013610","D"],"edges":["e-99e9156f8de29382","e-9995913d3601081a"],"text":"A304236 \u2192 basis A013610 \u2192 D"},"headline":"A304236 \u2192 basis A013610 \u2192 D","chain":null}