{"id":"n-8fb68f7b4056b77f","revision":0,"first_index":0,"fingerprint":"f6-fcecb9b7df4a0d68","name":"derivative of derivative of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","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":[[],[],["2"],["12","6"],["40","60","12"],["100","300","180","20"],["210","1050","1260","420","30"],["392","2940","5880","3920","840","42"],["672","7056","21168","23520","10080","1512","56"],["1080","15120","63504","105840","75600","22680","2520","72"],["1650","29700","166320","388080","415800","207900","46200","3960","90"],["2420","54450","392040","1219680","1829520","1372140","508200","87120","5940","110"],["3432","94380","849420","3397680","6795360","7135128","3963960","1132560","154440","8580","132"],["4732","156156","1717716","8588580","22084920","30918888","24048024","10306296","2342340","260260","12012","156"],["6370","248430","3279276","20040020","64414350","115945830","120240120","72144072","24594570","4554550","420420","16380","182"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","2","18","112","600","2970","14014","64064","286416","1259700","5471620","23535072","100440080","425904570"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["4"],["-4"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["2","-2"],["3","-5"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174607,"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-801d3796ac1bd68a"]},{"id":174608,"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-801d3796ac1bd68a"]},{"id":174609,"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-801d3796ac1bd68a"]}],"region_exact":{},"incoming_edges":[{"id":"e-801d3796ac1bd68a","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-8fb68f7b4056b77f","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":"e84261e205cb70e11be1610f0e9a5d9ca1bf54fdc259f92edc687f60c9c7f8d8","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-a889e9d86928450a","role":"sequence","name":"derivative of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","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-1838ebf0ab889b26","kind":"fingerprint_related","node_a_id":"n-8fb68f7b4056b77f","node_b_id":"n-e6d6a605a8a707e0","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-e6d6a605a8a707e0","other":{"id":"n-e6d6a605a8a707e0","name":"derivative of reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-fcecb9b7df4a0d68","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["12"],["64","32"]],"card_rows":[["12"],["64","32"],["200","300","60"],["480","1440","864","96"]],"row_sums_preview":["0","12","96","560","2880","13860","64064","288288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["reverse","D"],"edges":["e-8b10d4d15ed56acc","e-f06d6d4945c980f0"],"text":"A352880 \u2192 reverse \u2192 D"},"headline":"A352880 \u2192 reverse \u2192 D","chain":null}},{"id":"l-7374f0ffc37a3f97","kind":"fingerprint_related","node_a_id":"n-1dcf17454d9bc598","node_b_id":"n-8fb68f7b4056b77f","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-1dcf17454d9bc598","other":{"id":"n-1dcf17454d9bc598","name":"derivative of derivative of narayana","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":[[],[],["2"]],"card_rows":[["2"],["12","6"],["40","60","12"],["100","300","180","20"]],"row_sums_preview":["0","0","2","18","112","600","2970","14014"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["D","D"],"edges":["e-18f6a04857337ebb","e-980873aa4ad51a2c"],"text":"narayana \u2192 D \u2192 D"},"headline":"narayana \u2192 D \u2192 D","chain":null}},{"id":"l-94b189a4657fcb53","kind":"fingerprint_related","node_a_id":"n-b1a4ceaaf7992b60","node_b_id":"n-8fb68f7b4056b77f","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-f75c02c48c08a66d","kind":"fingerprint_related","node_a_id":"n-aa9a15392d2524ca","node_b_id":"n-8fb68f7b4056b77f","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-aa9a15392d2524ca","other":{"id":"n-aa9a15392d2524ca","name":"derivative of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-fcecb9b7df4a0d68","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8","4"]],"card_rows":[["2"],["8","4"],["20","30","6"],["40","120","72","8"]],"row_sums_preview":["0","2","12","56","240","990","4004","16016"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["D"],"edges":["e-0041de9aacba5ab3"],"text":"A108838 \u2192 D"},"headline":"A108838 \u2192 D","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:    0;\n  1:    0;\n  2:    2;\n  3:   12,     6;\n  4:   40,    60,    12;\n  5:  100,   300,   180,     20;\n  6:  210,  1050,  1260,    420,    30;\n  7:  392,  2940,  5880,   3920,   840,    42;\n  8:  672,  7056, 21168,  23520, 10080,  1512,   56;\n  9: 1080, 15120, 63504, 105840, 75600, 22680, 2520, 72;\n","recurrence_sentence":"The row polynomials satisfy (n^3-n^2-4*n+4)*P(n,x) = (n^3+3*n^2+2*n+(3*n^3-5*n^2-2*n)*x)*P(n-1,x) + (2*n^2*x-2*n^2*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 3*n^2 + 2*n + (3*n^3 - 5*n^2 - 2*n)*x)*P[n - 1, x] + (2*n^2*x - 2*n^2*x^2)*D[P[n - 1, x], x])/(n^3 - n^2 - 4*n + 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 + 3*n^2 + 2*n + (3*n^3 - 5*n^2 - 2*n)*x)*v[i - 1] + (2*n^2*x - 2*n^2*x^2)*deriv(v[i - 1], x))/(n^3 - n^2 - 4*n + 4)); 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-c2eaf5212a4dec13","steps":["D","D"],"edges":["e-1c13b2418122affc","e-801d3796ac1bd68a"],"text":"A001263 \u2192 D \u2192 D"},"headline":"A001263 \u2192 D \u2192 D","chain":null}