{"id":"n-56990ef96985b6f3","revision":0,"first_index":0,"fingerprint":"f6-71f59c90b30e1bfc","name":"one_plus_derivative of Triangle read by rows, (2, 3)-Lah numbers.","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"],["37","1"],["1864","102","1"],["122760","10264","203","1"],["10243872","1167424","35096","348","1"],["1056616992","154168416","6281112","94360","545","1"],["132034779648","23699253888","1229149056","24930512","217680","802","1"],["19658292890112","4221067880448","267863977728","6889088768","81001632","450464","1127","1"],["3438754166679552","865104623294976","65349982987776","2048444003840","30454998528","227644352","858704","1528","1"],["698322321862502400","202525804487374848","17845433463545856","663389285842944","11986378599680","113055612544","572410256","1534416","2013","1"],["162937067612317286400","53770975422562639872","5440700530846181376","235094016736198656","5023298474322944","57456506806144","366507350592","1317307424","2601720","2590","1"],["43288785311683697049600","16082401735747073507328","1845338210568368474112","91266238895785998336","2260499089587379200","30509146992483840","235611949105856","1065629173312","2819614424","4223560","3267","1"],["12991959034894446192230400","5385047382439251770769408","693534835207938505015296","38781468424507522203648","1096526088864723326976","17117708780335655424","155719446449781760","851548229949696","2832593752736","5680657664","6609064","4052","1"],["4373813345333027149932134400","2007171660852022342739558400","287658883722761840098934784","18005536954990842515619840","574164630442127801161728","10210288117980990818304","107246471413471843840","689901461858259456","2771842025898400","6983018744416","10871261544","10021544","4953","1"],["1641356255870968029464494080000","828446173822746247467486412800","131155896032898815989153529856","9112953792200325037225869312","324519612248052986982727680","6495040723226978919702528","77580456836922086354944","575437105869281538048","2716687586125407232","8259538279183360","16142976888896","19904229488","14787136","5978","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","38","1967","133228","11446741","1217161426","156988331587","24154195290280","4371287900469257","919369049107424958","222388918705270697719","61310082741811907012836","19110436521337622009956125","6687233909720513995334261610","2610402372330403604002930717259"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["37","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-500","155","-6","-1"],["-425","111","-6","0"],["-135","26","-1","0"],["-19","2","0","0"],["-1","0","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-220","89","-9"],["0","-139","38","-2"],["0","-29","4","0"],["0","-2","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-20","9","-1"],["0","0","-9","2","0"],["0","0","-1","0","0"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["56","-10","1"],["21","-2","0"],["2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","13"],["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194873,"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":true,"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-b9f8c313262da19c"]},{"id":194874,"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-b9f8c313262da19c"]},{"id":194875,"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-b9f8c313262da19c"]}],"region_exact":{},"incoming_edges":[{"id":"e-b9f8c313262da19c","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-56990ef96985b6f3","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":"7d36648f552b52d3ddc73fa05e95ec8b3a86eb35f982f362e5b53f7a4809421e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b8808e9caf8dcf35","role":"sequence","name":"Triangle read by rows, (2, 3)-Lah numbers.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"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:                  1;\n  1:                 37,                  1;\n  2:               1864,                102,                 1;\n  3:             122760,              10264,               203,               1;\n  4:           10243872,            1167424,             35096,             348,              1;\n  5:         1056616992,          154168416,           6281112,           94360,            545,            1;\n  6:       132034779648,        23699253888,        1229149056,        24930512,         217680,          802,         1;\n  7:     19658292890112,      4221067880448,      267863977728,      6889088768,       81001632,       450464,      1127,       1;\n  8:   3438754166679552,    865104623294976,    65349982987776,   2048444003840,    30454998528,    227644352,    858704,    1528,    1;\n  9: 698322321862502400, 202525804487374848, 17845433463545856, 663389285842944, 11986378599680, 113055612544, 572410256, 1534416, 2013, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4+19*n^3+135*n^2+425*n+500-(2*n^3+26*n^2+111*n+155)*x+(n^2+6*n+6)*x^2+x^3)*P(n-2,x) - ((2*n^3+29*n^2+139*n+220)*x-(4*n^2+38*n+89)*x^2+(2*n+9)*x^3)*d/dx P(n-2,x) - ((n^2+9*n+20)*x^2-(2*n+9)*x^3+x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+21*n+56-(2*n+10)*x+x^2)*P(n-1,x) + (2*n+13)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 37 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 37 + x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 + 19*n^3 + 135*n^2 + 425*n + 500 - (2*n^3 + 26*n^2 + 111*n + 155)*x + (n^2 + 6*n + 6)*x^2 + x^3)*P[n - 2, x] - ((2*n^3 + 29*n^2 + 139*n + 220)*x - (4*n^2 + 38*n + 89)*x^2 + (2*n + 9)*x^3)*D[P[n - 2, x], x] - ((n^2 + 9*n + 20)*x^2 - (2*n + 9)*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 21*n + 56 - (2*n + 10)*x + x^2)*P[n - 1, x] + (2*n + 13)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 37 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 19*n^3 + 135*n^2 + 425*n + 500 - (2*n^3 + 26*n^2 + 111*n + 155)*x + (n^2 + 6*n + 6)*x^2 + x^3)*v[i - 2] - ((2*n^3 + 29*n^2 + 139*n + 220)*x - (4*n^2 + 38*n + 89)*x^2 + (2*n + 9)*x^3)*deriv(v[i - 2], x) - ((n^2 + 9*n + 20)*x^2 - (2*n + 9)*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 21*n + 56 - (2*n + 10)*x + x^2)*v[i - 1] + (2*n + 13)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-b8808e9caf8dcf35","steps":["D+1"],"edges":["e-b9f8c313262da19c"],"text":"A371259 \u2192 D+1"},"headline":"A371259 \u2192 D+1","chain":null}