{"id":"n-fd9a7ad994e206d0","revision":0,"first_index":0,"fingerprint":"f6-7195245dea7f3067","name":"one_plus_derivative of basis_map_A007318 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","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"],["4","1"],["19","9","1"],["107","71","15","1"],["702","580","173","22","1"],["5274","5104","1910","345","30","1"],["44712","48860","21559","4915","610","39","1"],["422568","509004","255045","68929","10850","994","49","1"],["4407120","5753736","3193392","984704","184569","21532","1526","60","1"],["50292720","70290936","42461556","14562764","3126732","435981","39480","2238","72","1"],["623471040","924118272","599807916","224709140","53850635","8630433","937083","68070","3165","85","1"],["8344624320","13020978816","8991695196","3630938036","953466855","171076499","21412545","1870143","111705","4345","99","1"],["119938250880","195869441664","142799644008","61523748616","17462478286","3442630356","483682793","48820530","3514533","176000","5819","114","1"],["1842662908800","3134328981120","2397624269976","1093335698520","331921903742","70902233116","10971594062","1245985169","103891788","6283563","267982","7631","130","1"],["30136443724800","53180752331520","42469068928896","20365952993568","6558680696204","1501823106784","252353864763","31630994967","2973547148","208717509","10770760","396305","9828","147","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","29","194","1478","12664","120696","1267440","14546640","181212480","2435595840","35136178560","541572393600","8883104025600","154499900083200"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-2","-1"],["-2","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182755,"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-5aa3e7f816769045"]},{"id":182756,"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-5aa3e7f816769045"]},{"id":182757,"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-5aa3e7f816769045"]},{"id":182758,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-5aa3e7f816769045"]}],"region_exact":{},"incoming_edges":[{"id":"e-5aa3e7f816769045","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-fd9a7ad994e206d0","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":"317e2009844944196672f023e9b16114872a5225db7bde82ff2991e33a83d137","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-634bc9ca21d13c4b","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","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":[{"id":"l-86329aaa0b852b90","kind":"fingerprint_related","node_a_id":"n-fd9a7ad994e206d0","node_b_id":"n-b291e351cce4e0ed","fingerprint":"f6-7195245dea7f3067","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7195245dea7f3067"},"other_id":"n-b291e351cce4e0ed","other":{"id":"n-b291e351cce4e0ed","name":"one_plus_derivative of basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-7195245dea7f3067","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["19","9","1"],["107","71","15","1"],["702","580","173","22","1"]],"card_rows":[["19","9","1"],["107","71","15","1"],["702","580","173","22","1"],["5274","5104","1910","345","30","1"]],"row_sums_preview":["29","194","1478","12664","120696","1267440","14546640","181212480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":["basis A007318","D+1"],"edges":["e-a33c0ab38a356260","e-4f13f0918e896a8a"],"text":"A143491 \u2192 basis A007318 \u2192 D+1"},"headline":"A143491 \u2192 basis A007318 \u2192 D+1","chain":null}}],"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:        4,        1;\n  2:       19,        9,        1;\n  3:      107,       71,       15,        1;\n  4:      702,      580,      173,       22,       1;\n  5:     5274,     5104,     1910,      345,      30,      1;\n  6:    44712,    48860,    21559,     4915,     610,     39,     1;\n  7:   422568,   509004,   255045,    68929,   10850,    994,    49,    1;\n  8:  4407120,  5753736,  3193392,   984704,  184569,  21532,  1526,   60,  1;\n  9: 50292720, 70290936, 42461556, 14562764, 3126732, 435981, 39480, 2238, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(2*n+2)*x+x^2)*P(n-2,x) + (2*n+3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + 1 + (2*n + 2)*x + x^2)*P[n - 2, x] + (2*n + 3 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n + 1 + (2*n + 2)*x + x^2)*v[i - 2] + (2*n + 3 + 2*x)*v[i - 1]); 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-914700b1060feeb2","steps":["x \u21a6 x+1","D+1"],"edges":["e-10d14680c8c670e7","e-5aa3e7f816769045"],"text":"A136124 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A136124 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}
