{"id":"n-af7f1e73681bcc05","revision":1,"first_index":0,"fingerprint":"f6-df585348d4ef3b4c","name":"obreschkoff_combination of Unsigned 2-Stirling numbers of the first kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["30","31","10","1"],["144","180","80","15","1"],["840","1198","651","169","21","1"],["5760","9072","5684","1820","315","28","1"],["45360","77292","53964","20089","4320","538","36","1"],["403200","732960","557864","233100","58674","9135","861","45","1"],["3991680","7663536","6262740","2870440","815815","149513","17710","1310","55","1"],["43545600","87626880","76044672","37622904","11789756","2451834","343497","32076","1914","66","1"],["518918400","1088068320","994409208","524863756","178338810","41070875","6556914","727143","54990","2705","78","1"],["6706022400","14583179520","13945097088","7782670896","2832239696","709949240","125984144","15978963","1440153","90090","3718","91","1"],["93405312000","209863094400","208890420480","122412593016","47266716420","12732769126","2467496955","348993293","36087975","2698773","142065","4991","105","1"],["1394852659200","3227775989760","3330198422784","2037786541440","828748409840","237587064000","49625469608","7687113720","888925895","76482120","4826107","216840","6565","120","1"],["22230464256000","52842981139200","56315081312640","35819663267088","15253159311936","4618079854024","1029246834616","172285578465","21875227088","2110028492","153561408","8290646","321776","8484","136","1"],["376610217984000","917529654988800","1007065034472960","663326690130432","294339224947104","93534991346160","22067998316000","3950748265464","543310214018","57671840655","4715884745","294290451","13754377","465885","10795","153","1"],["6758061133824000","16842332515276800","18991519523170560","12913048460276352","5946963726489264","1973576514808144","489776991926040","93016534918944","13709311115643","1579367188113","142408892340","10005034156","541553922","22131942","660060","13548","171","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["72","420","2880","22680","201600","1995840","21772800","259459200","3353011200","46702656000","697426329600","11115232128000","188305108992000","3379030566912000","64023737057280000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["30","31","10","1"]],"recurrence":{"denominator":{"coeffs":[["4","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["15","8","1"],["8","2","0"],["1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162823,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-a410505258437c48"]}],"region_exact":{},"incoming_edges":[{"id":"e-a410505258437c48","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-af7f1e73681bcc05","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":"1a86566b333d91471cd35159a4cae5a63ac28ceb14e6cf3eea642e016e9df2a2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c237629cf5fdbb07","role":"sequence","name":"Unsigned 2-Stirling numbers of the first kind.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-c237629cf5fdbb07","role":"sequence","name":"Unsigned 2-Stirling numbers of the first kind.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-7646d852db87aff8","kind":"oeis_match","node_a_id":"n-af7f1e73681bcc05","node_b_id":null,"fingerprint":null,"oeis_id":"A171005","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":3,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A171005","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A171005","relation":"match_row_sums"}],"bfiles":[],"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:         30,          31,          10,          1;\n  1:        144,         180,          80,         15,          1;\n  2:        840,        1198,         651,        169,         21,         1;\n  3:       5760,        9072,        5684,       1820,        315,        28,         1;\n  4:      45360,       77292,       53964,      20089,       4320,       538,        36,        1;\n  5:     403200,      732960,      557864,     233100,      58674,      9135,       861,       45,       1;\n  6:    3991680,     7663536,     6262740,    2870440,     815815,    149513,     17710,     1310,      55,     1;\n  7:   43545600,    87626880,    76044672,   37622904,   11789756,   2451834,    343497,    32076,    1914,    66,    1;\n  8:  518918400,  1088068320,   994409208,  524863756,  178338810,  41070875,   6556914,   727143,   54990,  2705,   78,  1;\n  9: 6706022400, 14583179520, 13945097088, 7782670896, 2832239696, 709949240, 125984144, 15978963, 1440153, 90090, 3718, 91, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+4+x)*P(n,x) = (n^2+8*n+15+(2*n+8)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 30 + 31*x + 10*x^2 + x^3.","mathematica":"P[0, x_] := 30 + 31*x + 10*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 8*n + 15 + (2*n + 8)*x + x^2)*P[n - 1, x])/(n + 4 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 30 + 31*x + 10*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 8*n + 15 + (2*n + 8)*x + x^2)*v[i - 1])/(n + 4 + 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}}],"derivation":{"root":"n-c237629cf5fdbb07","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a410505258437c48"],"text":"A143491 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A143491 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}