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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"f1f2940d6f3905314369c888df639a553c23981ff545cc0870ab89cc559a862e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b0d6ec97a8d172b8","role":"sequence","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,        3;\n   3:      5,       24,        27;\n   4:     25,      171,       378,       270;\n   5:    129,     1152,      3780,      5400,       2835;\n   6:    681,     7515,     32670,     69930,      73710,      30618;\n   7:   3653,    48024,    259875,    740880,    1173690,     979776,     336798;\n   8:  19825,   302547,   1958796,   6974100,   14747670,   18523890,   12798324,    3752892;\n   9: 108545,  1886208,  14215824,  60691680,  160540380,  269438400,  280215936,  165127248,   42220035;\n  10: 598417, 11665971, 100312020, 499334220, 1585757250, 3331942614, 4632319692, 4109416740, 2111001750, 478493730;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(12*n-42)*x)*P(n-2,x) - (7*n-12+(12*n-18)*x)*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + 3*x, P(3,x) = 5 + 24*x + 27*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 3*x;\nP[3, x_] := 5 + 24*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*P[n - 3, x] + (7*n - 23 + (12*n - 42)*x)*P[n - 2, x] - (7*n - 12 + (12*n - 18)*x)*P[n - 1, x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; v[3] = 5 + 24*x + 27*x^2; for(n = 4, N, my(i = n); v[i] = (-(n - 4)*v[i - 3] + (7*n - 23 + (12*n - 42)*x)*v[i - 2] - (7*n - 12 + (12*n - 18)*x)*v[i - 1])/(-n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b0d6ec97a8d172b8","steps":["basis A013610"],"edges":["e-096f8d1cc00e8a39"],"text":"A253283 \u2192 basis A013610"},"headline":"A253283 \u2192 basis A013610","chain":null}