{"id":"n-0daa6c2f3598d022","revision":1,"first_index":0,"fingerprint":"f6-f490dd0849981934","name":"Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 7.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":true,"rows":[["1"],["-8","1"],["72","-17","1"],["-720","242","-27","1"],["7920","-3382","539","-38","1"],["-95040","48504","-9850","995","-50","1"],["1235520","-725592","176554","-22785","1645","-63","1"],["-17297280","11393808","-3197348","495544","-45815","2527","-77","1"],["259459200","-188204400","59354028","-10630508","1182769","-83720","3682","-92","1"],["-4151347200","3270729600","-1137868848","229442156","-29554812","2522289","-142632","5154","-108","1"],["70572902400","-59753750400","22614500016","-5038385500","731873960","-72433725","4947033","-230250","6990","-125","1"],["-1270312243200","1146140409600","-466814750688","113305439016","-18212116780","2035681010","-161480319","9091533","-356070","9240","-143","1"],["24135932620800","-23046980025600","10015620672672","-2619618091992","459335657836","-56890055970","5103807071","-334219446","15856863","-531630","11957","-162","1"],["-482718652416000","485075533132800","-223359393479040","62407982512512","-11806331248712","1597136777236","-158966197390","11788195991","-651356706","26489463","-770770","15197","-182","1"],["10137091700736000","-10669304848204800","5175622796192640","-1533927026241792","310340938735464","-45346203570668","4935426922426","-406518313201","25466686817","-1207635429","42675633","-1089907","19019","-203","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-7","56","-504","5040","-55440","665280","-8648640","121080960","-1816214400","29059430400","-494010316800","8892185702400","-168951528345600","3379030566912000"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A051379","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-7","1"],["-1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160058,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[15/2, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"15/2"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [15/2, +inf)","region_exact":null,"edges":["e-a85f747dc534a48f"]},{"id":160059,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-a85f747dc534a48f"]},{"id":160060,"position":2,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-a85f747dc534a48f"]}],"region_exact":{},"incoming_edges":[{"id":"e-a85f747dc534a48f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-2\",\"intercept\":\"15\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-0daa6c2f3598d022","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"ce883e52d2d33fd3f0d3e461b38e965c0b4dabc11ec75d80e30b1df04c252002","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"15/2"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[{"id":"e-646058a1a3e3d26a","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-645e3679f0201989","conclusion_name":"obreschkoff_combination of Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 7."},{"id":"e-646058a1a3e3d26a","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-645e3679f0201989","conclusion_name":"obreschkoff_combination of Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 7."}],"used_by_total":2,"links":[{"id":"l-6c497562c19ffffe","kind":"oeis_match","node_a_id":"n-0daa6c2f3598d022","node_b_id":null,"fingerprint":null,"oeis_id":"A051379","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A051379","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A051379","relation":"identification"},{"oeis_id":"A051379","relation":"match_triangle"}],"bfiles":[{"node_id":"n-0daa6c2f3598d022","oeis_id":"A051379","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":30181}],"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:          -8,          1;\n  2:          72,        -17,           1;\n  3:        -720,        242,         -27,         1;\n  4:        7920,      -3382,         539,       -38,         1;\n  5:      -95040,      48504,       -9850,       995,       -50,       1;\n  6:     1235520,    -725592,      176554,    -22785,      1645,     -63,       1;\n  7:   -17297280,   11393808,    -3197348,    495544,    -45815,    2527,     -77,    1;\n  8:   259459200, -188204400,    59354028, -10630508,   1182769,  -83720,    3682,  -92,    1;\n  9: -4151347200, 3270729600, -1137868848, 229442156, -29554812, 2522289, -142632, 5154, -108, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n+7-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n + 7 - x)*P[n - 1, 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] = -(n + 7 - x)*v[i - 1]); 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":"A051379","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:          -8,          1;\n  2:          72,        -17,           1;\n  3:        -720,        242,         -27,         1;\n  4:        7920,      -3382,         539,       -38,         1;\n  5:      -95040,      48504,       -9850,       995,       -50,       1;\n  6:     1235520,    -725592,      176554,    -22785,      1645,     -63,       1;\n  7:   -17297280,   11393808,    -3197348,    495544,    -45815,    2527,     -77,    1;\n  8:   259459200, -188204400,    59354028, -10630508,   1182769,  -83720,    3682,  -92,    1;\n  9: -4151347200, 3270729600, -1137868848, 229442156, -29554812, 2522289, -142632, 5154, -108, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n+7-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n + 7 - x)*P[n - 1, 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] = -(n + 7 - x)*v[i - 1]); 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}}],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 \u22122x+15"],"edges":["e-a85f747dc534a48f"],"text":"A028338 \u2192 x \u21a6 \u22122x+15"},"headline":"A051379: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 7.","chain":"A028338 \u2192 x \u21a6 \u22122x+15"}