{"id":"n-880485a065802fcb","revision":1,"first_index":0,"fingerprint":"f6-d26c9cf8645537ad","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 = 2.","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"],["-3","1"],["12","-7","1"],["-60","47","-12","1"],["360","-342","119","-18","1"],["-2520","2754","-1175","245","-25","1"],["20160","-24552","12154","-3135","445","-33","1"],["-181440","241128","-133938","40369","-7140","742","-42","1"],["1814400","-2592720","1580508","-537628","111769","-14560","1162","-52","1"],["-19958400","30334320","-19978308","7494416","-1767087","271929","-27342","1734","-63","1"],["239500800","-383970240","270074016","-109911300","28699460","-5030235","600033","-48150","2490","-75","1"],["-3113510400","5231113920","-3894932448","1698920916","-483004280","94092515","-12830664","1225983","-80520","3465","-88","1"],["43589145600","-76349105280","59760168192","-27679825272","8460980836","-1800299490","273721811","-29994426","2353263","-129030","4697","-102","1"],["-653837184000","1188825724800","-972751628160","474957547272","-154594537812","35465473186","-5906126655","723638201","-65293371","4288713","-199485","6227","-117","1"],["10461394944000","-19675048780800","16752851775360","-8572072384512","2948470152264","-722042108788","129963499666","-17484337871","1768332137","-133912779","7480473","-299117","8099","-133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-2","6","-24","120","-720","5040","-40320","362880","-3628800","39916800","-479001600","6227020800","-87178291200","1307674368000"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A049458","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","1"],["-1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160043,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[5/2, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"5/2"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [5/2, +inf)","region_exact":null,"edges":["e-88d2cf9ed2610e27"]},{"id":160044,"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-88d2cf9ed2610e27"]},{"id":160045,"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-88d2cf9ed2610e27"]}],"region_exact":{},"incoming_edges":[{"id":"e-88d2cf9ed2610e27","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-2\",\"intercept\":\"5\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-880485a065802fcb","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":"ec8f8e94fe949ab1a29497160a54c861a22da25922b94f634831a95e3a90009d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"5/2"}}},{"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-6ddcf5c6f38774e7","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-954f0c79bf5b2ac0","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 = 2."},{"id":"e-6ddcf5c6f38774e7","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-954f0c79bf5b2ac0","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 = 2."}],"used_by_total":2,"links":[{"id":"l-fd17fa1f7b1700c1","kind":"oeis_match","node_a_id":"n-880485a065802fcb","node_b_id":null,"fingerprint":null,"oeis_id":"A049458","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":40},"in_catalog":true,"oeis_id":"A049458","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-ff1fce9759990374","kind":"oeis_match","node_a_id":"n-880485a065802fcb","node_b_id":null,"fingerprint":null,"oeis_id":"A155456","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A155456","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A049458","relation":"identification"},{"oeis_id":"A155456","relation":"match_row_sums"},{"oeis_id":"A049458","relation":"match_triangle"}],"bfiles":[{"node_id":"n-880485a065802fcb","oeis_id":"A049458","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":40,"oeis_terms_agreeing":40,"oeis_terms_available":40,"term_count":990,"raw_size":28369}],"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:        -3,        1;\n  2:        12,       -7,         1;\n  3:       -60,       47,       -12,       1;\n  4:       360,     -342,       119,     -18,        1;\n  5:     -2520,     2754,     -1175,     245,      -25,      1;\n  6:     20160,   -24552,     12154,   -3135,      445,    -33,      1;\n  7:   -181440,   241128,   -133938,   40369,    -7140,    742,    -42,    1;\n  8:   1814400, -2592720,   1580508, -537628,   111769, -14560,   1162,  -52,   1;\n  9: -19958400, 30334320, -19978308, 7494416, -1767087, 271929, -27342, 1734, -63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n+2-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 + 2 - 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 + 2 - 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":"A049458","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,        1;\n  2:        12,       -7,         1;\n  3:       -60,       47,       -12,       1;\n  4:       360,     -342,       119,     -18,        1;\n  5:     -2520,     2754,     -1175,     245,      -25,      1;\n  6:     20160,   -24552,     12154,   -3135,      445,    -33,      1;\n  7:   -181440,   241128,   -133938,   40369,    -7140,    742,    -42,    1;\n  8:   1814400, -2592720,   1580508, -537628,   111769, -14560,   1162,  -52,   1;\n  9: -19958400, 30334320, -19978308, 7494416, -1767087, 271929, -27342, 1734, -63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n+2-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 + 2 - 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 + 2 - 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+5"],"edges":["e-88d2cf9ed2610e27"],"text":"A028338 \u2192 x \u21a6 \u22122x+5"},"headline":"A049458: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 2.","chain":"A028338 \u2192 x \u21a6 \u22122x+5"}
