{"id":"n-af0fa7591e93aaa0","revision":0,"first_index":0,"fingerprint":"f6-e8b03ce9b132c5b4","name":"one_plus_derivative of Delannoy paths counted by number of weak peaks.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["9","16","4"],["19","90","60","8"],["33","320","528","192","16"],["51","850","2800","2400","560","32"],["73","1872","10500","17600","9360","1536","64"],["99","3626","31164","88200","90160","32928","4032","128"],["129","6400","78400","338688","580160","401408","107520","10240","256"],["163","10530","174528","1072512","2794176","3217536","1612800","331776","25344","512"],["201","16400","353700","2937600","10866240","18966528","15724800","5990400","979200","61440","1024"],["243","24442","665500","7187400","35893440","88792704","111234816","69696000","20908800","2787840","146432","2048"],["289","35136","1179024","16068800","104282640","347922432","616670208","582100992","285753600","69386240","7704576","344064","4096"],["339","49010","1987440","33372768","273198640","1183751712","2826869760","3769159680","2779755264","1099338240","220819456","20766720","798720","8192"],["393","66640","3213028","65188032","657312656","3591171584","11130799680","19976546304","20777492736","12312588288","4010918912","678379520","54792192","1835008","16384"],["451","88650","5012700","120902600","1472693040","9907785888","38705867200","90106473600","126031276800","105170024960","51209174016","13991577600","2022092800","141926400","4177920","32768"],["513","115712","7584000","214502400","3105706240","25235718144","121366941696","355976192000","645468595200","723678658560","495488401408","201885745152","46964736000","5872025600","361758720","9437184","65536"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["29","177","1089","6693","41005","250337","1523201","9239877","55897533","337339665","2031452097","12209875941","73260321357","438889107393","2625636184065"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["9","16","4"]],"recurrence":{"denominator":{"coeffs":[["6","2","4"],["11","9","2"],["8","4","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["18","18","28","16"],["25","65","52","12"],["12","52","24","0"],["2","12","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","12","-14","-16","-8"],["0","14","-20","-16","0"],["0","4","-8","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183336,"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-ec79ff25696a2a92"]},{"id":183337,"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-ec79ff25696a2a92"]},{"id":183338,"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-ec79ff25696a2a92"]},{"id":183339,"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-ec79ff25696a2a92"]}],"region_exact":{},"incoming_edges":[{"id":"e-ec79ff25696a2a92","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-af0fa7591e93aaa0","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":"57c7ced3cb0a7cabf01d29e7e2ad0054795332cdf85d85f699af0a55694fa295","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-b31aa1a3fd93d174","role":"sequence","name":"Delannoy paths counted by number of weak peaks.","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:   9,    16,      4;\n  1:  19,    90,     60,       8;\n  2:  33,   320,    528,     192,       16;\n  3:  51,   850,   2800,    2400,      560,       32;\n  4:  73,  1872,  10500,   17600,     9360,     1536,        64;\n  5:  99,  3626,  31164,   88200,    90160,    32928,      4032,      128;\n  6: 129,  6400,  78400,  338688,   580160,   401408,    107520,    10240,      256;\n  7: 163, 10530, 174528, 1072512,  2794176,  3217536,   1612800,   331776,    25344,     512;\n  8: 201, 16400, 353700, 2937600, 10866240, 18966528,  15724800,  5990400,   979200,   61440,   1024;\n  9: 243, 24442, 665500, 7187400, 35893440, 88792704, 111234816, 69696000, 20908800, 2787840, 146432, 2048;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+8*n^2+11*n+6+(4*n^2+9*n+2)*x+(2*n+4)*x^2)*P(n,x) = (2*n^3+12*n^2+25*n+18+(12*n^3+52*n^2+65*n+18)*x+(24*n^2+52*n+28)*x^2+(12*n+16)*x^3)*P(n-1,x) + ((4*n^2+14*n+12)*x-(8*n^2+20*n+14)*x^2-(16*n+16)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 9 + 16*x + 4*x^2.","mathematica":"P[0, x_] := 9 + 16*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 12*n^2 + 25*n + 18 + (12*n^3 + 52*n^2 + 65*n + 18)*x + (24*n^2 + 52*n + 28)*x^2 + (12*n + 16)*x^3)*P[n - 1, x] + ((4*n^2 + 14*n + 12)*x - (8*n^2 + 20*n + 14)*x^2 - (16*n + 16)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 8*n^2 + 11*n + 6 + (4*n^2 + 9*n + 2)*x + (2*n + 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 9 + 16*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + 12*n^2 + 25*n + 18 + (12*n^3 + 52*n^2 + 65*n + 18)*x + (24*n^2 + 52*n + 28)*x^2 + (12*n + 16)*x^3)*v[i - 1] + ((4*n^2 + 14*n + 12)*x - (8*n^2 + 20*n + 14)*x^2 - (16*n + 16)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 8*n^2 + 11*n + 6 + (4*n^2 + 9*n + 2)*x + (2*n + 4)*x^2)); 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-b31aa1a3fd93d174","steps":["D+1"],"edges":["e-ec79ff25696a2a92"],"text":"A133214 \u2192 D+1"},"headline":"A133214 \u2192 D+1","chain":null}