{"id":"n-d04cf1089b783b9c","revision":0,"first_index":1,"fingerprint":"f6-ced16c5d17801fb2","name":"one_plus_derivative of Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..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"],["2","1"],["7","6","1"],["34","39","12","1"],["209","292","126","20","1"],["1546","2505","1360","310","30","1"],["13327","24306","15675","4580","645","42","1"],["130922","263431","194796","68635","12530","1197","56","1"],["1441729","3154824","2612092","1068536","237510","29624","2044","72","1"],["17572114","41368977","37705536","17453100","4541796","694134","62832","3276","90","1"],["234662231","589410910","583900245","300216120","89050710","16024932","1785210","122520","4995","110","1"],["3405357682","9064804551","9664662700","5443344885","1805152140","372185814","49016352","4153050","223410","7315","132","1"],["53334454417","149641946796","170364779706","103978789420","37991860335","8798594904","1332567852","133864632","8910495","385660","10362","156","1"],["896324308634","2638693215769","3187414790352","2089971641186","831825249970","213160570623","36387004368","4215009084","333366462","17884295","636064","14274","182","1"],["16083557845279","49490245341642","63094659575311","44134205331580","18960454243731","5313358688638","1006792725939","131784135912","12043396365","768950182","33942909","1009372","19201","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","14","86","648","5752","58576","671568","8546432","119401856","1815177984","29808908032","525586164736","9898343691264","198227905206272"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181564,"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-271e415dbcb3a2ae"]},{"id":181565,"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-271e415dbcb3a2ae"]},{"id":181566,"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-271e415dbcb3a2ae"]}],"region_exact":{},"incoming_edges":[{"id":"e-271e415dbcb3a2ae","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-d04cf1089b783b9c","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"4c9d221a473952a033ed0e65ca78b8c8acb94aadaca6b23b392d7527d85b9a04","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-01897e65d9bb3867","role":"sequence","name":"Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..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-2ce9344088890c18","kind":"fingerprint_related","node_a_id":"n-d04cf1089b783b9c","node_b_id":"n-18383b729ceef751","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-18383b729ceef751","other":{"id":"n-18383b729ceef751","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","first_index":1,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["4","24","28"],["8","96","312","272"],["16","320","2016","4672","3344"]],"card_rows":[["4","24","28"],["8","96","312","272"],["16","320","2016","4672","3344"],["32","960","9920","43520","80160","49472"]],"row_sums_preview":["56","688","10368","184064","3748864","85960704","2187886592","61133750272"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":["x \u21a6 x/(x+1)"],"edges":["e-c6b10d6190de873e"],"text":"A395972 \u2192 x \u21a6 x/(x+1)"},"headline":"A395972 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-60b65a0845696217","kind":"fingerprint_related","node_a_id":"n-d04cf1089b783b9c","node_b_id":"n-184f8d63c8db0227","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-184f8d63c8db0227","other":{"id":"n-184f8d63c8db0227","name":"shift_x_plus_one of Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.","first_index":1,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1","-2"],["1","6","7"]],"card_rows":[["1"],["-1","-2"],["1","6","7"],["-1","-12","-39","-34"]],"row_sums_preview":["1","-3","14","-86","648","-5752","58576","-671568"],"regions":["(-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-db839974123087ee","steps":["x \u21a6 x+1"],"edges":["e-f386eb84f6c9d238"],"text":"A123202 \u2192 x \u21a6 x+1"},"headline":"A123202 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-6325edfaec3db966","kind":"fingerprint_related","node_a_id":"n-148a4e95b74cc241","node_b_id":"n-d04cf1089b783b9c","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-148a4e95b74cc241","other":{"id":"n-148a4e95b74cc241","name":"mobius_x_over_x_plus_one of T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","first_index":0,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","7"]],"card_rows":[["1"],["1","2"],["1","6","7"],["1","12","39","34"]],"row_sums_preview":["1","3","14","86","648","5752","58576","671568"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-47fe395e9f7d5f86","steps":["x \u21a6 x/(x+1)"],"edges":["e-6a2c9a503d8f313d"],"text":"A144084 \u2192 x \u21a6 x/(x+1)"},"headline":"A144084 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7bbeaeccdab2f637","kind":"fingerprint_related","node_a_id":"n-c0db4c865e1a9a1a","node_b_id":"n-d04cf1089b783b9c","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-c0db4c865e1a9a1a","other":{"id":"n-c0db4c865e1a9a1a","name":"derivative of Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","first_index":1,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","2"]],"card_rows":[["1"],["4","2"],["21","18","3"],["136","156","48","4"]],"row_sums_preview":["0","1","6","42","344","3240","34512","410032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01897e65d9bb3867","steps":["D"],"edges":["e-71ee257a063243ca"],"text":"A059110 \u2192 D"},"headline":"A059110 \u2192 D","chain":null}},{"id":"l-cc42b48a88c2f03f","kind":"fingerprint_related","node_a_id":"n-b9a589cc34ffc178","node_b_id":"n-d04cf1089b783b9c","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-b9a589cc34ffc178","other":{"id":"n-b9a589cc34ffc178","name":"shift_x_plus_one of basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["7","6","1"]],"card_rows":[["1"],["2","1"],["7","6","1"],["34","39","12","1"]],"row_sums_preview":["1","3","14","86","648","5752","58576","671568"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-07cfebc08bea7e3c","e-8357b32e6e3ab54f"],"text":"A132393 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A132393 \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:        2,        1;\n   3:        7,        6,        1;\n   4:       34,       39,       12,        1;\n   5:      209,      292,      126,       20,       1;\n   6:     1546,     2505,     1360,      310,      30,      1;\n   7:    13327,    24306,    15675,     4580,     645,     42,     1;\n   8:   130922,   263431,   194796,    68635,   12530,   1197,    56,    1;\n   9:  1441729,  3154824,  2612092,  1068536,  237510,  29624,  2044,   72,  1;\n  10: 17572114, 41368977, 37705536, 17453100, 4541796, 694134, 62832, 3276, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+x)*P(n-1,x) + (1+x)*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[(n + x)*P[n - 1, x] + (1 + x)*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] = (n + x)*v[i - 1] + (1 + x)*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-01897e65d9bb3867","steps":["D+1"],"edges":["e-271e415dbcb3a2ae"],"text":"A059110 \u2192 D+1"},"headline":"A059110 \u2192 D+1","chain":null}
