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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":"d5c5de3c4ca454161ac57c85dd387d13f1362500a0a0b91398b36e9aa06641b5","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":[],"used_by_total":0,"links":[{"id":"l-a743659f82b12d45","kind":"oeis_match","node_a_id":"n-3f596f3a08c8e69f","node_b_id":null,"fingerprint":null,"oeis_id":"A146988","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":"A146988","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a943b77762a3a510","kind":"oeis_match","node_a_id":"n-3f596f3a08c8e69f","node_b_id":null,"fingerprint":null,"oeis_id":"A146988","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":"A146988","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A146988","relation":"identification"},{"oeis_id":"A146988","relation":"match_reversed_triangle"},{"oeis_id":"A146988","relation":"match_triangle"}],"bfiles":[{"node_id":"n-3f596f3a08c8e69f","oeis_id":"A146988","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":25354},{"node_id":"n-3f596f3a08c8e69f","oeis_id":"A146988","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":25354}],"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,     6,      1;\n  3: 1,    19,     19,       1;\n  4: 1,    68,    134,      68,       1;\n  5: 1,   261,    778,     778,     261,       1;\n  6: 1,  1030,   4111,    6164,    4111,    1030,       1;\n  7: 1,  4103,  20501,   40995,   40995,   20501,    4103,      1;\n  8: 1, 16392,  98332,  245816,  327750,  245816,   98332,  16392,     1;\n  9: 1, 65545, 458788, 1376340, 2293886, 2293886, 1376340, 458788, 65545, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-1,x) - (4*n-12+(8*n-24)*x+(4*n-12)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 6*x + x^2, P(3,x) = 1 + 19*x + 19*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 6*x + x^2;\nP[3, x_] := 1 + 19*x + 19*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 1, x] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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 + 6*x + x^2; v[4] = 1 + 19*x + 19*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 1] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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":"A146988","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,     6,      1;\n  3: 1,    19,     19,       1;\n  4: 1,    68,    134,      68,       1;\n  5: 1,   261,    778,     778,     261,       1;\n  6: 1,  1030,   4111,    6164,    4111,    1030,       1;\n  7: 1,  4103,  20501,   40995,   40995,   20501,    4103,      1;\n  8: 1, 16392,  98332,  245816,  327750,  245816,   98332,  16392,     1;\n  9: 1, 65545, 458788, 1376340, 2293886, 2293886, 1376340, 458788, 65545, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-1,x) - (4*n-12+(8*n-24)*x+(4*n-12)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 6*x + x^2, P(3,x) = 1 + 19*x + 19*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 6*x + x^2;\nP[3, x_] := 1 + 19*x + 19*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 1, x] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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 + 6*x + x^2; v[4] = 1 + 19*x + 19*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 1] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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":"A146988","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,     6,      1;\n  3: 1,    19,     19,       1;\n  4: 1,    68,    134,      68,       1;\n  5: 1,   261,    778,     778,     261,       1;\n  6: 1,  1030,   4111,    6164,    4111,    1030,       1;\n  7: 1,  4103,  20501,   40995,   40995,   20501,    4103,      1;\n  8: 1, 16392,  98332,  245816,  327750,  245816,   98332,  16392,     1;\n  9: 1, 65545, 458788, 1376340, 2293886, 2293886, 1376340, 458788, 65545, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-1,x) - (4*n-12+(8*n-24)*x+(4*n-12)*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 6*x + x^2, P(3,x) = 1 + 19*x + 19*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 6*x + x^2;\nP[3, x_] := 1 + 19*x + 19*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 1, x] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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 + 6*x + x^2; v[4] = 1 + 19*x + 19*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 1] - (4*n - 12 + (8*n - 24)*x + (4*n - 12)*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-3f596f3a08c8e69f","steps":[],"edges":[],"text":"A146988"},"headline":"A146988: Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 4^(n-1) * binomial(n-2, k-1) otherwise.","chain":null}