{"id":"n-aa9a15392d2524ca","revision":1,"first_index":0,"fingerprint":"f6-fcecb9b7df4a0d68","name":"derivative of Triangle of Dyck paths counted by number of long interior inclines.","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":[[],["2"],["8","4"],["20","30","6"],["40","120","72","8"],["70","350","420","140","10"],["112","840","1680","1120","240","12"],["168","1764","5292","5880","2520","378","14"],["240","3360","14112","23520","16800","5040","560","16"],["330","5940","33264","77616","83160","41580","9240","792","18"],["440","9900","71280","221760","332640","249480","92400","15840","1080","20"],["572","15730","141570","566280","1132560","1189188","660660","188760","25740","1430","22"],["728","24024","264264","1321320","3397680","4756752","3699696","1585584","360360","40040","1848","24"],["910","35490","468468","2862860","9202050","16563690","17177160","10306296","3513510","650650","60060","2340","26"],["1120","50960","794976","5829824","22902880","51531480","68708640","54966912","26234208","7287280","1121120","87360","2912","28"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","12","56","240","990","4004","16016","63648","251940","994840","3922512","15452320","60843510","239519700"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-3"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["6","-4"],["5","1"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164318,"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-0041de9aacba5ab3"]},{"id":164319,"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-0041de9aacba5ab3"]},{"id":164320,"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-0041de9aacba5ab3"]},{"id":164321,"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-0041de9aacba5ab3"]}],"region_exact":{},"incoming_edges":[{"id":"e-0041de9aacba5ab3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-aa9a15392d2524ca","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":"c34c431556a1e3a5ffee0464844bc76e4940554dc8c3b454fb0469d93e338744","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-c963abfc0266d66c","role":"sequence","name":"Triangle of Dyck paths counted by number of long interior inclines.","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-d48f2fde9c8db626","kind":"fingerprint_related","node_a_id":"n-aa9a15392d2524ca","node_b_id":"n-b1a4ceaaf7992b60","fingerprint":"f6-fcecb9b7df4a0d68","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fcecb9b7df4a0d68"},"other_id":"n-b1a4ceaaf7992b60","other":{"id":"n-b1a4ceaaf7992b60","name":"derivative of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","first_index":1,"fingerprint":"f6-fcecb9b7df4a0d68","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["10","20"]],"card_rows":[["4"],["10","20"],["18","90","60"],["28","252","420","140"]],"row_sums_preview":["0","4","30","168","840","3960","18018","80080"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2304b34e998691c","steps":["D"],"edges":["e-36b6691ae4af4747"],"text":"A062145 \u2192 D"},"headline":"A062145 \u2192 D","chain":null}},{"id":"l-99cef6a201bcd7b5","kind":"oeis_match","node_a_id":"n-aa9a15392d2524ca","node_b_id":null,"fingerprint":null,"oeis_id":"A006659","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A006659","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A006659","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-aa9a15392d2524ca","oeis_id":"A006659","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":13032}],"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:   0;\n  1:   2;\n  2:   8,    4;\n  3:  20,   30,     6;\n  4:  40,  120,    72,     8;\n  5:  70,  350,   420,   140,    10;\n  6: 112,  840,  1680,  1120,   240,    12;\n  7: 168, 1764,  5292,  5880,  2520,   378,   14;\n  8: 240, 3360, 14112, 23520, 16800,  5040,  560,  16;\n  9: 330, 5940, 33264, 77616, 83160, 41580, 9240, 792, 18;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n-3)*P(n,x) = (n^2+5*n+6+(3*n^2+n-4)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 6 + (3*n^2 + n - 4)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 6 + (3*n^2 + n - 4)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n - 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-c963abfc0266d66c","steps":["D"],"edges":["e-0041de9aacba5ab3"],"text":"A108838 \u2192 D"},"headline":"A108838 \u2192 D","chain":null}