{"id":"n-19ac71afae259650","revision":1,"first_index":1,"fingerprint":"f6-4ba9b6f073520673","name":"obreschkoff_combination of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","1"],["0","2","2"],["0","2","8","6"],["0","2","20","42","24"],["0","2","44","186","264","120"],["0","2","92","690","1800","1920","720"],["0","2","188","2346","9960","18600","15840","5040"],["0","2","380","7602","49224","142800","206640","146160","40320"],["0","2","764","23946","227304","960120","2096640","2469600","1491840","362880"],["0","2","1532","74130","1005000","5937120","18340560","31963680","31691520","16692480","3628800"],["0","2","3068","226986","4316520","34710600","145666080","352129680","509241600","435456000","203212800","39916800"],["0","2","6140","690162","18174024","195135600","1082260080","3484570320","6890970240","8502278400","6386688000","2674425600","479001600"],["0","2","12284","2088906","75456744","1066548120","7664374080","31967812800","83004324480","138539237760","148889664000","99672249600","37841126400","6227020800"],["0","2","24572","6303570","310182600","5710024320","52385533200","277425308160","919777098240","1993892060160","2874289017600","2734181049600","1650160512000","572885913600","87178291200"],["0","2","49148","18984426","1265944680","30101034600","348573345120","2308675889520","9577619251200","26223022425600","48681810777600","61693170739200","52612098739200","28899603532800","9240898867200","1307674368000"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["2","4","16","88","616","5224","51976","593128","7633096","109334824","1724880136","29714200168","554949915976","11168201318824","240924533948296"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162743,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-db08ce0645ad4c2a"]}],"region_exact":{},"incoming_edges":[{"id":"e-db08ce0645ad4c2a","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-19ac71afae259650","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"77fe476d9d34ee7027582ed3fa29f705d244d04544769f24e36306005aa79507","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-2cabfd060b04f15d","role":"sequence","name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-2cabfd060b04f15d","role":"sequence","name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-036dea8512d6a4da","kind":"fingerprint_related","node_a_id":"n-19ac71afae259650","node_b_id":"n-46dff17fbd7d3a46","fingerprint":"f6-4ba9b6f073520673","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4ba9b6f073520673"},"other_id":"n-46dff17fbd7d3a46","other":{"id":"n-46dff17fbd7d3a46","name":"obreschkoff_combination of Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).","first_index":1,"fingerprint":"f6-4ba9b6f073520673","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2","2"],["2","8","6"],["2","20","42","24"]],"card_rows":[["2","2"],["2","8","6"],["2","20","42","24"],["2","44","186","264","120"]],"row_sums_preview":["4","16","88","616","5224","51976","593128","7633096"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d938c1937a93c4df","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-d58e3f8b704ee8a6"],"text":"A019538 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A019538 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}}],"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, 1;\n   2: 0, 2,    2;\n   3: 0, 2,    8,     6;\n   4: 0, 2,   20,    42,      24;\n   5: 0, 2,   44,   186,     264,     120;\n   6: 0, 2,   92,   690,    1800,    1920,      720;\n   7: 0, 2,  188,  2346,    9960,   18600,    15840,     5040;\n   8: 0, 2,  380,  7602,   49224,  142800,   206640,   146160,    40320;\n   9: 0, 2,  764, 23946,  227304,  960120,  2096640,  2469600,  1491840,   362880;\n  10: 0, 2, 1532, 74130, 1005000, 5937120, 18340560, 31963680, 31691520, 16692480, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + x.","mathematica":"P[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + (x + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2cabfd060b04f15d","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-db08ce0645ad4c2a"],"text":"A131689 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A131689 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}