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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":"13ea7d410571f8ca82fc20b2b4604675074e3d71f69c8787356305162b20cf76","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0eda1df9a0edca27","kind":"oeis_match","node_a_id":"n-55c53861db6c014a","node_b_id":null,"fingerprint":null,"oeis_id":"A350512","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A350512","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4aafe9173f76b502","kind":"oeis_match","node_a_id":"n-55c53861db6c014a","node_b_id":null,"fingerprint":null,"oeis_id":"A057711","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A057711","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-614c86c3f1a68c31","kind":"oeis_match","node_a_id":"n-55c53861db6c014a","node_b_id":null,"fingerprint":null,"oeis_id":"A129952","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A129952","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7333edcf628b7f3c","kind":"oeis_match","node_a_id":"n-55c53861db6c014a","node_b_id":null,"fingerprint":null,"oeis_id":"A350512","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"reversed_triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A350512","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A350512","relation":"identification"},{"oeis_id":"A350512","relation":"match_reversed_triangle"},{"oeis_id":"A057711","relation":"match_row_sums"},{"oeis_id":"A129952","relation":"match_row_sums"},{"oeis_id":"A350512","relation":"match_triangle"}],"bfiles":[{"node_id":"n-55c53861db6c014a","oeis_id":"A350512","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":11543},{"node_id":"n-55c53861db6c014a","oeis_id":"A350512","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":11543}],"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,  4,   1;\n  3: 1,  7,   7,   1;\n  4: 1, 10,  18,  10,   1;\n  5: 1, 13,  34,  34,  13,   1;\n  6: 1, 16,  55,  80,  55,  16,   1;\n  7: 1, 19,  81, 155, 155,  81,  19,   1;\n  8: 1, 22, 112, 266, 350, 266, 112,  22,  1;\n  9: 1, 25, 148, 420, 686, 686, 420, 148, 25, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 4*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n - 2)]];\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 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n - 2)); 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":"A350512","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,  4,   1;\n  3: 1,  7,   7,   1;\n  4: 1, 10,  18,  10,   1;\n  5: 1, 13,  34,  34,  13,   1;\n  6: 1, 16,  55,  80,  55,  16,   1;\n  7: 1, 19,  81, 155, 155,  81,  19,   1;\n  8: 1, 22, 112, 266, 350, 266, 112,  22,  1;\n  9: 1, 25, 148, 420, 686, 686, 420, 148, 25, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 4*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n - 2)]];\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 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n - 2)); 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":"A350512","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,  4,   1;\n  3: 1,  7,   7,   1;\n  4: 1, 10,  18,  10,   1;\n  5: 1, 13,  34,  34,  13,   1;\n  6: 1, 16,  55,  80,  55,  16,   1;\n  7: 1, 19,  81, 155, 155,  81,  19,   1;\n  8: 1, 22, 112, 266, 350, 266, 112,  22,  1;\n  9: 1, 25, 148, 420, 686, 686, 420, 148, 25, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 4*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n - 2)]];\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 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n - 2)); 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-55c53861db6c014a","steps":[],"edges":[],"text":"A350512"},"headline":"A350512: Triangle read by rows with T(n,0) = 1 for n >= 0 and T(n,k) = binomial(n-1,k-1)*(2*k*(n-k) + n)/k for 0 < k <= n.","chain":null}