{"id":"n-3c9d2e7be059a416","revision":0,"first_index":0,"fingerprint":"f6-1f9cbdd862ac2135","name":"one_plus_derivative of reversal of Triangular array formed by the Mersenne 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"],["4","3"],["7","20","7"],["10","55","68","15"],["13","108","264","196","31"],["16","179","676","1002","516","63"],["19","268","1385","3216","3309","1284","127"],["22","375","2472","7945","12910","9969","3076","255"],["25","500","4018","16620","37680","46140","28150","7172","511"],["28","643","6104","30996","91056","156414","151544","75772","16388","1023"],["31","804","8811","53152","192766","431256","588182","466848","196611","36868","2047"],["34","983","12220","85491","370044","1026606","1830416","2049310","1367858","495627","81924","4095"],["37","1180","16412","130740","658845","2189880","4858896","7130696","6719935","3849900","1220628","180228","8191"],["40","1395","21468","191950","1105060","4290345","11440920","20951316","25928496","20975525","10485260","2949150","393220","16383"],["43","1628","27469","272496","1765731","7853868","24531573","54129984","83765001","89090676","62854911","27786576","7012393","851972","32767"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","34","148","612","2452","9608","37024","140816","529968","1977376","7324608","26965568","98750528","359977088"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0"],["4","-24","4","2"],["-14","18","4","0"],["12","2","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","23","-7","-7","-1"],["-2","-42","-99","11","11","1"],["-2","-24","81","27","2","0"],["12","56","15","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","2","7","-29","-11","-1"],["0","-2","-23","-70","-23","-2","0"],["0","-6","-37","-12","-1","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174516,"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-c3806f15d6a8f3d0"]},{"id":174517,"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-c3806f15d6a8f3d0"]},{"id":174518,"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-c3806f15d6a8f3d0"]}],"region_exact":{},"incoming_edges":[{"id":"e-c3806f15d6a8f3d0","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-3c9d2e7be059a416","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":"52bf0005e4dd688218bb3a8294e1174b4c9e90705f75e0cffb551455f2db7a67","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-369f42b9099fbd84","role":"sequence","name":"reversal of Triangular array formed by the Mersenne 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:  4,   3;\n  2:  7,  20,    7;\n  3: 10,  55,   68,    15;\n  4: 13, 108,  264,   196,    31;\n  5: 16, 179,  676,  1002,   516,     63;\n  6: 19, 268, 1385,  3216,  3309,   1284,    127;\n  7: 22, 375, 2472,  7945, 12910,   9969,   3076,   255;\n  8: 25, 500, 4018, 16620, 37680,  46140,  28150,  7172,   511;\n  9: 28, 643, 6104, 30996, 91056, 156414, 151544, 75772, 16388, 1023;\n","recurrence_sentence":"The row polynomials satisfy (12*n^3-14*n^2+4*n+(2*n^3+18*n^2-24*n)*x+(4*n^2+4*n)*x^2+2*n*x^3)*P(n,x) = (12*n^3-2*n^2-2*n+(56*n^3-24*n^2-42*n)*x+(15*n^3+81*n^2-99*n+23)*x^2+(n^3+27*n^2+11*n-7)*x^3+(2*n^2+11*n-7)*x^4+(n-1)*x^5)*P(n-1,x) - ((6*n^2+2*n)*x+(37*n^2+23*n-2)*x^2+(12*n^2+70*n-7)*x^3+(n^2+23*n+29)*x^4+(2*n+11)*x^5+x^6)*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[Together[((12*n^3 - 2*n^2 - 2*n + (56*n^3 - 24*n^2 - 42*n)*x + (15*n^3 + 81*n^2 - 99*n + 23)*x^2 + (n^3 + 27*n^2 + 11*n - 7)*x^3 + (2*n^2 + 11*n - 7)*x^4 + (n - 1)*x^5)*P[n - 1, x] - ((6*n^2 + 2*n)*x + (37*n^2 + 23*n - 2)*x^2 + (12*n^2 + 70*n - 7)*x^3 + (n^2 + 23*n + 29)*x^4 + (2*n + 11)*x^5 + x^6)*D[P[n - 1, x], x])/(12*n^3 - 14*n^2 + 4*n + (2*n^3 + 18*n^2 - 24*n)*x + (4*n^2 + 4*n)*x^2 + 2*n*x^3)]];\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] = ((12*n^3 - 2*n^2 - 2*n + (56*n^3 - 24*n^2 - 42*n)*x + (15*n^3 + 81*n^2 - 99*n + 23)*x^2 + (n^3 + 27*n^2 + 11*n - 7)*x^3 + (2*n^2 + 11*n - 7)*x^4 + (n - 1)*x^5)*v[i - 1] - ((6*n^2 + 2*n)*x + (37*n^2 + 23*n - 2)*x^2 + (12*n^2 + 70*n - 7)*x^3 + (n^2 + 23*n + 29)*x^4 + (2*n + 11)*x^5 + x^6)*deriv(v[i - 1], x))/(12*n^3 - 14*n^2 + 4*n + (2*n^3 + 18*n^2 - 24*n)*x + (4*n^2 + 4*n)*x^2 + 2*n*x^3)); 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-9b5c7e97e5191da6","steps":["reverse","D+1"],"edges":["e-5faf1c18373ba45e","e-c3806f15d6a8f3d0"],"text":"A110441 \u2192 reverse \u2192 D+1"},"headline":"A110441 \u2192 reverse \u2192 D+1","chain":null}