{"id":"n-86fb6463ff8368f2","revision":1,"first_index":0,"fingerprint":"f6-0a12f7fa088d0629","name":"Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","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","3"],["1","12","15"],["1","33","141","105"],["1","78","786","1830","945"],["1","171","3450","17538","26685","10395"],["1","360","13257","125352","396495","435960","135135"],["1","741","46971","753291","4238811","9356175","7921305","2027025"],["1","1506","157956","4046526","37013166","140913270","233216460","158799690","34459425"],["1","3039","513024","20135004","282182454","1696782438","4732607160","6167943180","3486128625","654729075"],["1","6108","1627203","94904688","1954557378","17517438432","75547811070","162926117280","173237850765","83242120500","13749310575"],["1","12249","5077065","430062045","12619927530","161786794554","1019322953586","3343199837130","5795219706165","5163367474125","2149053308325","316234143225"],["1","24534","15659910","1892868390","77291685135","1374558448284","12150651306276","57325287304620","149109772632255","213899826513870","163050508192950","59670195735150","7905853580625"],["1","49107","47912022","8150890230","454601687715","10952559669429","131789546385588","859573791544068","3176397147438135","6761401216738605","8210550385538550","5444507086417350","1773541577132325","213458046676875"],["1","98256","145749453","34520041800","2590893157545","82990222149744","1327771532467989","11621014002233712","58672657030985595","175611515180282640","313442294393298015","328071697374241800","191835760181658075","56194659967446000","6190283353629375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","28","280","3640","58240","1106560","24344320","608608000","17041024000","528271744000","17961239296000","664565853952000","26582634158080000","1143053268797440000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A219512","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","3"]]},"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":159620,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-5be8dcded4b783ba"]},{"id":159621,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-5be8dcded4b783ba"]},{"id":159622,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-5be8dcded4b783ba"]},{"id":159623,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-5be8dcded4b783ba"]}],"region_exact":{},"incoming_edges":[{"id":"e-5be8dcded4b783ba","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-86fb6463ff8368f2","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"49fc42cb2738fcb3636dada5c03d892d6fee02d093502c541c5a5778e0931211","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-b4f1d8e4113d9eff","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-f055c1e733379fba","conclusion_name":"obreschkoff_combination of Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents."},{"id":"e-b4f1d8e4113d9eff","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-f055c1e733379fba","conclusion_name":"obreschkoff_combination of Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents."}],"used_by_total":2,"links":[{"id":"l-1273d120b1ec7c1a","kind":"oeis_match","node_a_id":"n-86fb6463ff8368f2","node_b_id":null,"fingerprint":null,"oeis_id":"A007559","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":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007559","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-87970a9042495cb5","kind":"oeis_match","node_a_id":"n-86fb6463ff8368f2","node_b_id":null,"fingerprint":null,"oeis_id":"A219512","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A219512","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A219512","relation":"identification"},{"oeis_id":"A007559","relation":"match_row_sums"},{"oeis_id":"A219512","relation":"match_triangle"}],"bfiles":[{"node_id":"n-86fb6463ff8368f2","oeis_id":"A219512","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":40913}],"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: 1,    3;\n  2: 1,   12,     15;\n  3: 1,   33,    141,      105;\n  4: 1,   78,    786,     1830,       945;\n  5: 1,  171,   3450,    17538,     26685,      10395;\n  6: 1,  360,  13257,   125352,    396495,     435960,     135135;\n  7: 1,  741,  46971,   753291,   4238811,    9356175,    7921305,    2027025;\n  8: 1, 1506, 157956,  4046526,  37013166,  140913270,  233216460,  158799690,   34459425;\n  9: 1, 3039, 513024, 20135004, 282182454, 1696782438, 4732607160, 6167943180, 3486128625, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], 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] = (1 + 3*n*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A219512","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    3;\n   3: 1,   12,     15;\n   4: 1,   33,    141,      105;\n   5: 1,   78,    786,     1830,       945;\n   6: 1,  171,   3450,    17538,     26685,      10395;\n   7: 1,  360,  13257,   125352,    396495,     435960,     135135;\n   8: 1,  741,  46971,   753291,   4238811,    9356175,    7921305,    2027025;\n   9: 1, 1506, 157956,  4046526,  37013166,  140913270,  233216460,  158799690,   34459425;\n  10: 1, 3039, 513024, 20135004, 282182454, 1696782438, 4732607160, 6167943180, 3486128625, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(3*n-3)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (3*n - 3)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (3*n - 3)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-86fb6463ff8368f2","steps":[],"edges":[],"text":"A219512"},"headline":"A219512: Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","chain":null}