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Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"8c1043d612b8aa9ec528b106dc96a8f83d5b1beefd1708998de24f1c1c99cb81","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":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-9e5f717e87031ade","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-14aaf5fc5867d5df","conclusion_name":"gamma_polynomial of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 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begins:\n  0:     1;\n  1:     3,        3;\n  2:     9,       42,          9;\n  3:    27,      393,        393,          27;\n  4:    81,     3156,       8646,        3156,           81;\n  5:   243,    23631,     142446,      142446,        23631,          243;\n  6:   729,   171006,    2015895,     4273380,      2015895,       171006,         729;\n  7:  2187,  1216725,   26107983,   102402705,    102402705,     26107983,     1216725,       2187;\n  8:  6561,  8584872,  320039388,  2136524184,   3891302790,   2136524184,   320039388,    8584872,     6561;\n  9: 19683, 60323739, 3786564300, 40688926236, 123074809242, 123074809242, 40688926236, 3786564300, 60323739, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(4*n-1)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + (4*n - 1)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + (4*n - 1)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x)); 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":"A257621","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:     3,        3;\n  2:     9,       42,          9;\n  3:    27,      393,        393,          27;\n  4:    81,     3156,       8646,        3156,           81;\n  5:   243,    23631,     142446,      142446,        23631,          243;\n  6:   729,   171006,    2015895,     4273380,      2015895,       171006,         729;\n  7:  2187,  1216725,   26107983,   102402705,    102402705,     26107983,     1216725,       2187;\n  8:  6561,  8584872,  320039388,  2136524184,   3891302790,   2136524184,   320039388,    8584872,     6561;\n  9: 19683, 60323739, 3786564300, 40688926236, 123074809242, 123074809242, 40688926236, 3786564300, 60323739, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(4*n-1)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + (4*n - 1)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + (4*n - 1)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x)); 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":"A257621","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:     3,        3;\n  2:     9,       42,          9;\n  3:    27,      393,        393,          27;\n  4:    81,     3156,       8646,        3156,           81;\n  5:   243,    23631,     142446,      142446,        23631,          243;\n  6:   729,   171006,    2015895,     4273380,      2015895,       171006,         729;\n  7:  2187,  1216725,   26107983,   102402705,    102402705,     26107983,     1216725,       2187;\n  8:  6561,  8584872,  320039388,  2136524184,   3891302790,   2136524184,   320039388,    8584872,     6561;\n  9: 19683, 60323739, 3786564300, 40688926236, 123074809242, 123074809242, 40688926236, 3786564300, 60323739, 19683;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+(4*n-1)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + (4*n - 1)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x]];\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; for(n = 1, N, my(i = n + 1); v[i] = (3 + (4*n - 1)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x)); 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-930be05ae3b321d9","steps":[],"edges":[],"text":"A257621"},"headline":"A257621: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.","chain":null}