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L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"halfGrowth","lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"certificate_hash":"08f05f3ec1d55852d8fe6f6781b25df962f569ab647418d0d186ad719408d632","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-9b31724594a81849","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-4aa60654988fe331","conclusion_name":"gamma_polynomial of Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows."}],"used_by_total":1,"links":[{"id":"l-39bb89e1c9790054","kind":"oeis_match","node_a_id":"n-14be0ce6edbde2a1","node_b_id":null,"fingerprint":null,"oeis_id":"A144439","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":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A144439","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f240644b06b62cab","kind":"oeis_match","node_a_id":"n-14be0ce6edbde2a1","node_b_id":null,"fingerprint":null,"oeis_id":"A144439","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":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A144439","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A144439","relation":"identification"},{"oeis_id":"A144439","relation":"match_reversed_triangle"},{"oeis_id":"A144439","relation":"match_triangle"}],"bfiles":[{"node_id":"n-14be0ce6edbde2a1","oeis_id":"A144439","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":35449},{"node_id":"n-14be0ce6edbde2a1","oeis_id":"A144439","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":35449}],"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,     8,       1;\n  3: 1,    31,      31,        1;\n  4: 1,   102,     326,      102,         1;\n  5: 1,   317,    2406,     2406,       317,         1;\n  6: 1,   964,   15087,    34336,     15087,       964,        1;\n  7: 1,  2907,   86673,   380947,    380947,     86673,     2907,       1;\n  8: 1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;\n  9: 1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n-1)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n - 1)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x] + 2*x*P[n - 2, x]];\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; for(n = 2, N, my(i = n + 1); v[i] = (1 + (2*n - 1)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x) + 2*x*v[i - 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":"A144439","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,     1;\n   3: 1,     8,       1;\n   4: 1,    31,      31,        1;\n   5: 1,   102,     326,      102,         1;\n   6: 1,   317,    2406,     2406,       317,         1;\n   7: 1,   964,   15087,    34336,     15087,       964,        1;\n   8: 1,  2907,   86673,   380947,    380947,     86673,     2907,       1;\n   9: 1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;\n  10: 1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n-3)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) + 2*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n - 3)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x] + 2*x*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (2*n - 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x) + 2*x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A144439","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,     1;\n   3: 1,     8,       1;\n   4: 1,    31,      31,        1;\n   5: 1,   102,     326,      102,         1;\n   6: 1,   317,    2406,     2406,       317,         1;\n   7: 1,   964,   15087,    34336,     15087,       964,        1;\n   8: 1,  2907,   86673,   380947,    380947,     86673,     2907,       1;\n   9: 1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;\n  10: 1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(2*n-3)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) + 2*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n - 3)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x] + 2*x*P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (2*n - 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x) + 2*x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-14be0ce6edbde2a1","steps":[],"edges":[],"text":"A144439"},"headline":"A144439: Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.","chain":null}