{"id":"n-72fe1904e30238b0","revision":0,"first_index":1,"fingerprint":"f6-1090d35600282d8c","name":"reversal of shift_x_plus_one of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","5","5"],["1","9","22","15"],["1","14","61","99","52"],["1","20","135","385","471","203"],["1","27","260","1140","2416","2386","877"],["1","35","455","2835","9156","15470","12867","4140"],["1","44","742","6230","28441","72590","102215","73681","21147"],["1","54","1146","12474","76503","274386","578929","700311","446620","115975"],["1","65","1695","23220","184317","884751","2607760","4692500","4984651","2856457","678570"],["1","77","2420","40755","406857","2521827","9906919","24762815","38878191","36872561","19217243","4213597"],["1","90","3355","68145","836187","6509580","32929534","109242210","236989676","330314776","283342060","135610448","27644437"],["1","104","4537","109395","1619332","15484755","98205250","418245542","1199262350","2298463882","2882874644","2259923016","1001159901","190899322"],["1","119","6006","169624","2981979","34391357","267626788","1425970832","5232663436","13202773584","22669969686","25868151764","18689475897","7714225057","1382958545"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","47","227","1215","7107","44959","305091","2206399","16913987","136823263","1163490499","10366252031","96491364675"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0","-2"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177489,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-9dcad2a28def6e29"]},{"id":177490,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-9dcad2a28def6e29"]},{"id":177491,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-9dcad2a28def6e29"]}],"region_exact":{},"incoming_edges":[{"id":"e-9dcad2a28def6e29","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-72fe1904e30238b0","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"2283ac255a6949f42e04d1e4fca1a10ba70d9c3a27e266013fa0e8e8d0a3e69f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-45544f9d4b113e1b","role":"sequence","name":"shift_x_plus_one of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-47b166cc8439cad1","kind":"candidate_equal","node_a_id":"n-185306a9f278f7c5","node_b_id":"n-72fe1904e30238b0","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-185306a9f278f7c5","other":{"id":"n-185306a9f278f7c5","name":"mobius_x_over_x_plus_one of Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-1090d35600282d8c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","9","22","15"]],"row_sums_preview":["1","3","11","47","227","1215","7107","44959"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f277d949034445ba","steps":["x \u21a6 x/(x+1)"],"edges":["e-556cd21d36039ca7"],"text":"A008278 \u2192 x \u21a6 x/(x+1)"},"headline":"A008278 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-204b27a04f886f52","kind":"fingerprint_related","node_a_id":"n-bdae5aa3d1cac0ec","node_b_id":"n-72fe1904e30238b0","fingerprint":"f6-1090d35600282d8c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1090d35600282d8c"},"other_id":"n-bdae5aa3d1cac0ec","other":{"id":"n-bdae5aa3d1cac0ec","name":"Triangle read by rows: A008277 * A007318.","first_index":0,"fingerprint":"f6-1090d35600282d8c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["15","22","9","1"]],"row_sums_preview":["1","3","11","47","227","1215","7107","44959"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":["A137597"],"derivation":{"root":"n-bdae5aa3d1cac0ec","steps":[],"edges":[],"text":"A137597"},"headline":"A137597: Triangle read by rows: A008277 * A007318.","chain":null}},{"id":"l-22fa39edc554d646","kind":"fingerprint_related","node_a_id":"n-2e7e4df17772fe4d","node_b_id":"n-72fe1904e30238b0","fingerprint":"f6-1090d35600282d8c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1090d35600282d8c"},"other_id":"n-2e7e4df17772fe4d","other":{"id":"n-2e7e4df17772fe4d","name":"reversal of Triangle read by rows: A008277 * A007318.","first_index":0,"fingerprint":"f6-1090d35600282d8c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","9","22","15"]],"row_sums_preview":["1","3","11","47","227","1215","7107","44959"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bdae5aa3d1cac0ec","steps":["reverse"],"edges":["e-475fdd3e596ddb02"],"text":"A137597 \u2192 reverse"},"headline":"A137597 \u2192 reverse","chain":null}},{"id":"l-247fb43d9f055ee0","kind":"fingerprint_related","node_a_id":"n-72fe1904e30238b0","node_b_id":"n-867753e2092cf52b","fingerprint":"f6-1090d35600282d8c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1090d35600282d8c"},"other_id":"n-867753e2092cf52b","other":{"id":"n-867753e2092cf52b","name":"shift_x_plus_one of basis_map_A008277 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1090d35600282d8c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["20","20","4"]],"card_rows":[["1"],["4","2"],["20","20","4"],["120","176","72","8"]],"row_sums_preview":["1","6","44","376","3632","38880","454848","5754752"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-dbb6853ea73a13a7","e-340d61d5be3ee8de"],"text":"A130123 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A130123 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-75211f1a857c845a","kind":"fingerprint_related","node_a_id":"n-72fe1904e30238b0","node_b_id":"n-f2565c43d8698f03","fingerprint":"f6-1090d35600282d8c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1090d35600282d8c"},"other_id":"n-f2565c43d8698f03","other":{"id":"n-f2565c43d8698f03","name":"shift_x_plus_one of basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-1090d35600282d8c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["5","5","1"],["15","22","9","1"],["52","99","61","14","1"]],"card_rows":[["5","5","1"],["15","22","9","1"],["52","99","61","14","1"],["203","471","385","135","20","1"]],"row_sums_preview":["11","47","227","1215","7107","44959","305091","2206399"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-66762082773dd3e0","e-cba3345425ab69e4"],"text":"basis-map coefficients of A112227 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A112227 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-9fe9957efb2bfb96","kind":"fingerprint_related","node_a_id":"n-45544f9d4b113e1b","node_b_id":"n-72fe1904e30238b0","fingerprint":"f6-1090d35600282d8c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1090d35600282d8c"},"other_id":"n-45544f9d4b113e1b","other":{"id":"n-45544f9d4b113e1b","name":"shift_x_plus_one of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-1090d35600282d8c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["15","22","9","1"]],"row_sums_preview":["1","3","11","47","227","1215","7107","44959"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["x \u21a6 x+1"],"edges":["e-656506c9153d3a60"],"text":"A008277 \u2192 x \u21a6 x+1"},"headline":"A008277 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-ebbd9e14d2dfe652","kind":"fingerprint_related","node_a_id":"n-72fe1904e30238b0","node_b_id":"n-3e5803db2c95b6c4","fingerprint":"f6-1090d35600282d8c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1090d35600282d8c"},"other_id":"n-3e5803db2c95b6c4","other":{"id":"n-3e5803db2c95b6c4","name":"shift_x_plus_one of basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-1090d35600282d8c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-2","-1"],["10","10","2"]],"card_rows":[["1"],["-2","-1"],["10","10","2"],["-90","-132","-54","-6"]],"row_sums_preview":["1","-3","22","-282","5448","-145800","5117040","-226593360"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-2b559797932c7395","e-cdcfc2c38c309600"],"text":"basis-map coefficients of A021012 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A021012 \u2192 basis A008277 \u2192 x \u21a6 x+1","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;\n   2: 1,  2;\n   3: 1,  5,    5;\n   4: 1,  9,   22,    15;\n   5: 1, 14,   61,    99,    52;\n   6: 1, 20,  135,   385,   471,    203;\n   7: 1, 27,  260,  1140,  2416,   2386,    877;\n   8: 1, 35,  455,  2835,  9156,  15470,  12867,   4140;\n   9: 1, 44,  742,  6230, 28441,  72590, 102215,  73681,  21147;\n  10: 1, 54, 1146, 12474, 76503, 274386, 578929, 700311, 446620, 115975;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n*x+(n-2)*x^2)*P(n-1,x) - (x^2+x^3)*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 + n*x + (n - 2)*x^2)*P[n - 1, x] - (x^2 + x^3)*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; for(n = 2, N, my(i = n); v[i] = (1 + n*x + (n - 2)*x^2)*v[i - 1] - (x^2 + x^3)*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-3cb22164e2f83fe1","steps":["x \u21a6 x+1","reverse"],"edges":["e-656506c9153d3a60","e-9dcad2a28def6e29"],"text":"A008277 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A008277 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}