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Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"eedaa5c32bac793b8b5e2af55adac2ea11b11ad19f5dc7a19cbdf0bdac096347","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-0bc3868d003d16cc","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-6587bc79a3c1f503","conclusion_name":"basis_map_A007318 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise."},{"id":"e-1d9d834688f391fd","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-919aa614b1337522","conclusion_name":"basis_map_A008277 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise."},{"id":"e-5c3f56110d60bf0e","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-c22b94fee16d0774","conclusion_name":"basis_map_A013610 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise."},{"id":"e-6feb4d060c383157","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-3f2d3a36a2f92f4a","conclusion_name":"basis_map_A013611 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise."},{"id":"e-d7b86a8b5889d427","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-9addcdd37c167f65","conclusion_name":"basis_map_A013609 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise."}],"used_by_total":5,"links":[{"id":"l-4252b387df00100d","kind":"oeis_match","node_a_id":"n-fbab854f2622405b","node_b_id":null,"fingerprint":null,"oeis_id":"A146987","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A146987","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b40e96d2037b0038","kind":"oeis_match","node_a_id":"n-fbab854f2622405b","node_b_id":null,"fingerprint":null,"oeis_id":"A146987","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A146987","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A146987","relation":"identification"},{"oeis_id":"A146987","relation":"match_reversed_triangle"},{"oeis_id":"A146987","relation":"match_triangle"}],"bfiles":[{"node_id":"n-fbab854f2622405b","oeis_id":"A146987","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":22148},{"node_id":"n-fbab854f2622405b","oeis_id":"A146987","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":22148}],"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: 1,    1;\n  2: 1,    5,     1;\n  3: 1,   12,    12,      1;\n  4: 1,   31,    60,     31,      1;\n  5: 1,   86,   253,    253,     86,      1;\n  6: 1,  249,   987,   1478,    987,    249,      1;\n  7: 1,  736,  3666,   7325,   7325,   3666,    736,     1;\n  8: 1, 2195, 13150,  32861,  43810,  32861,  13150,  2195,    1;\n  9: 1, 6570, 45963, 137865, 229761, 229761, 137865, 45963, 6570, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (4*n-12+(4*n-12)*x)*P(n-1,x) - (3*n-9+(6*n-18)*x+(3*n-9)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2, P(3,x) = 1 + 12*x + 12*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 5*x + x^2;\nP[3, x_] := 1 + 12*x + 12*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 12 + (4*n - 12)*x)*P[n - 1, x] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*P[n - 2, x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 5*x + x^2; v[4] = 1 + 12*x + 12*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((4*n - 12 + (4*n - 12)*x)*v[i - 1] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*v[i - 2])/(n - 3)); 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}},{"oeis_id":"A146987","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,    1;\n  2: 1,    5,     1;\n  3: 1,   12,    12,      1;\n  4: 1,   31,    60,     31,      1;\n  5: 1,   86,   253,    253,     86,      1;\n  6: 1,  249,   987,   1478,    987,    249,      1;\n  7: 1,  736,  3666,   7325,   7325,   3666,    736,     1;\n  8: 1, 2195, 13150,  32861,  43810,  32861,  13150,  2195,    1;\n  9: 1, 6570, 45963, 137865, 229761, 229761, 137865, 45963, 6570, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (4*n-12+(4*n-12)*x)*P(n-1,x) - (3*n-9+(6*n-18)*x+(3*n-9)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2, P(3,x) = 1 + 12*x + 12*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 5*x + x^2;\nP[3, x_] := 1 + 12*x + 12*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 12 + (4*n - 12)*x)*P[n - 1, x] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*P[n - 2, x])/(n - 3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 5*x + x^2; v[4] = 1 + 12*x + 12*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((4*n - 12 + (4*n - 12)*x)*v[i - 1] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*v[i - 2])/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}},{"oeis_id":"A146987","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,    1;\n  2: 1,    5,     1;\n  3: 1,   12,    12,      1;\n  4: 1,   31,    60,     31,      1;\n  5: 1,   86,   253,    253,     86,      1;\n  6: 1,  249,   987,   1478,    987,    249,      1;\n  7: 1,  736,  3666,   7325,   7325,   3666,    736,     1;\n  8: 1, 2195, 13150,  32861,  43810,  32861,  13150,  2195,    1;\n  9: 1, 6570, 45963, 137865, 229761, 229761, 137865, 45963, 6570, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (4*n-12+(4*n-12)*x)*P(n-1,x) - (3*n-9+(6*n-18)*x+(3*n-9)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2, P(3,x) = 1 + 12*x + 12*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 5*x + x^2;\nP[3, x_] := 1 + 12*x + 12*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 12 + (4*n - 12)*x)*P[n - 1, x] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*P[n - 2, x])/(n - 3)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 5*x + x^2; v[4] = 1 + 12*x + 12*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((4*n - 12 + (4*n - 12)*x)*v[i - 1] - (3*n - 9 + (6*n - 18)*x + (3*n - 9)*x^2)*v[i - 2])/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-fbab854f2622405b","steps":[],"edges":[],"text":"A146987"},"headline":"A146987: Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.","chain":null}