{"id":"n-6724d1657fe35b01","revision":0,"first_index":1,"fingerprint":"f6-b29d29aac1e7588a","name":"theta_plus_one of derivative of narayana","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":[[],["1"],["3","4"],["6","24","9"],["10","80","90","16"],["15","200","450","240","25"],["21","420","1575","1680","525","36"],["28","784","4410","7840","4900","1008","49"],["36","1344","10584","28224","29400","12096","1764","64"],["45","2160","22680","84672","132300","90720","26460","2880","81"],["55","3300","44550","221760","485100","498960","242550","52800","4455","100"],["66","4840","81675","522720","1524600","2195424","1600830","580800","98010","6600","121"],["78","6864","141570","1132560","4247100","8154432","8324316","4530240","1274130","171600","9438","144"],["91","9464","234234","2290288","10735725","26501904","36072036","27483456","11594583","2602600","286286","13104","169"],["105","12740","372645","4372368","25050025","77297220","135270135","137417280","81162081","27327300","5010005","458640","17745","196"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","7","39","196","930","4257","19019","83512","361998","1553630","6615686","27992472","117823940","493768485"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2","-4"],["-1","4"],["1","3"],["0","-4"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","4","2"],["1","-12","-5"],["1","5","14"],["0","-2","-12"],["0","1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","0","1"],["0","2","0","-2"],["0","0","-4","4"],["0","0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174496,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-32471eefdd663a56"]},{"id":174497,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-32471eefdd663a56"]},{"id":174498,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-32471eefdd663a56"]}],"region_exact":{},"incoming_edges":[{"id":"e-32471eefdd663a56","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-6724d1657fe35b01","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"8307e103b4393523823af3c6cb2f90d059772b341b023db057d67c2f0a3cf5f6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-28f1bce6fca8e347","role":"sequence","name":"derivative of narayana","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1f04c52fa360b323","kind":"fingerprint_related","node_a_id":"n-6724d1657fe35b01","node_b_id":"n-31c316014331006b","fingerprint":"f6-b29d29aac1e7588a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b29d29aac1e7588a"},"other_id":"n-31c316014331006b","other":{"id":"n-31c316014331006b","name":"theta_plus_one 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-b29d29aac1e7588a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["18","24"],["32","128","48"]],"card_rows":[["8"],["18","24"],["32","128","48"],["50","400","450","80"]],"row_sums_preview":["8","42","208","980","4464","19866","86944","375804"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["reverse","\u03b8+1"],"edges":["e-8b10d4d15ed56acc","e-4f9d3d3faef075e6"],"text":"A352880 \u2192 reverse \u2192 \u03b8+1"},"headline":"A352880 \u2192 reverse \u2192 \u03b8+1","chain":null}},{"id":"l-58d28a918db0700a","kind":"fingerprint_related","node_a_id":"n-a5bcbfbd7519fc4f","node_b_id":"n-6724d1657fe35b01","fingerprint":"f6-b29d29aac1e7588a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b29d29aac1e7588a"},"other_id":"n-a5bcbfbd7519fc4f","other":{"id":"n-a5bcbfbd7519fc4f","name":"theta_plus_one of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-b29d29aac1e7588a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["3","4"],["4","16","6"]],"card_rows":[["2"],["3","4"],["4","16","6"],["5","40","45","8"]],"row_sums_preview":["2","7","26","98","372","1419","5434","20878"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["\u03b8+1"],"edges":["e-12dbe0b95f16e416"],"text":"A108838 \u2192 \u03b8+1"},"headline":"A108838 \u2192 \u03b8+1","chain":null}},{"id":"l-d7e49479ce4e3879","kind":"fingerprint_related","node_a_id":"n-6724d1657fe35b01","node_b_id":"n-736ab6b73f0e9cf3","fingerprint":"f6-b29d29aac1e7588a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b29d29aac1e7588a"},"other_id":"n-736ab6b73f0e9cf3","other":{"id":"n-736ab6b73f0e9cf3","name":"theta_plus_one 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.","first_index":0,"fingerprint":"f6-b29d29aac1e7588a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","4"]],"card_rows":[["1"],["3","4"],["6","24","9"],["10","80","90","16"]],"row_sums_preview":["0","1","7","39","196","930","4257","19019"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["D","\u03b8+1"],"edges":["e-1c13b2418122affc","e-817facf47c189749"],"text":"A001263 \u2192 D \u2192 \u03b8+1"},"headline":"A001263 \u2192 D \u2192 \u03b8+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  0;\n   2:  1;\n   3:  3,    4;\n   4:  6,   24,     9;\n   5: 10,   80,    90,    16;\n   6: 15,  200,   450,   240,     25;\n   7: 21,  420,  1575,  1680,    525,    36;\n   8: 28,  784,  4410,  7840,   4900,  1008,    49;\n   9: 36, 1344, 10584, 28224,  29400, 12096,  1764,   64;\n  10: 45, 2160, 22680, 84672, 132300, 90720, 26460, 2880, 81;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-2+(n^4-4*n^3+3*n^2+4*n-4)*x)*P(n,x) = (n^2+n+(n^4-2*n^3+5*n^2-12*n+4)*x+(3*n^4-12*n^3+14*n^2-5*n+2)*x^2)*P(n-1,x) + ((2*n-1)*x+(2*n^3-4*n^2)*x^2-(2*n^3-4*n^2+2*n-1)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (n^4 - 2*n^3 + 5*n^2 - 12*n + 4)*x + (3*n^4 - 12*n^3 + 14*n^2 - 5*n + 2)*x^2)*P[n - 1, x] + ((2*n - 1)*x + (2*n^3 - 4*n^2)*x^2 - (2*n^3 - 4*n^2 + 2*n - 1)*x^3)*D[P[n - 1, x], x])/(n^2 - n - 2 + (n^4 - 4*n^3 + 3*n^2 + 4*n - 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 + n + (n^4 - 2*n^3 + 5*n^2 - 12*n + 4)*x + (3*n^4 - 12*n^3 + 14*n^2 - 5*n + 2)*x^2)*v[i - 1] + ((2*n - 1)*x + (2*n^3 - 4*n^2)*x^2 - (2*n^3 - 4*n^2 + 2*n - 1)*x^3)*deriv(v[i - 1], x))/(n^2 - n - 2 + (n^4 - 4*n^3 + 3*n^2 + 4*n - 4)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2a05d8340c329b21","steps":["D","\u03b8+1"],"edges":["e-18f6a04857337ebb","e-32471eefdd663a56"],"text":"narayana \u2192 D \u2192 \u03b8+1"},"headline":"narayana \u2192 D \u2192 \u03b8+1","chain":null}