{"id":"n-8552357c29e7b09e","revision":1,"first_index":0,"fingerprint":"f6-4e2cf4a8bda0ea40","name":"derivative of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","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"],["9","6"],["24","48","12"],["50","200","150","20"],["90","600","900","360","30"],["147","1470","3675","2940","735","42"],["224","3136","11760","15680","7840","1344","56"],["324","6048","31752","63504","52920","18144","2268","72"],["450","10800","75600","211680","264600","151200","37800","3600","90"],["605","18150","163350","609840","1067220","914760","381150","72600","5445","110"],["792","29040","326700","1568160","3659040","4390848","2744280","871200","130680","7920","132"],["1014","44616","613470","3680820","11042460","17667936","15459444","7361640","1840410","223080","11154","156"],["1274","66248","1093092","8016008","30060030","61837776","72144072","48096048","18036018","3643640","364364","15288","182"],["1575","95550","1863225","16396380","75150075","193243050","289864575","257657400","135270135","40990950","6831825","573300","20475","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","15","84","420","1980","9009","40040","175032","755820","3233230","13728792","57946200","243374040","1017958725"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"],["9","6"]],"recurrence":{"denominator":{"coeffs":[["0"],["4"],["-10"],["2"],["4"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["-22","54","-42","10"],["22","-54","42","-10"],["52","-108","60","-4"],["-52","108","-60","4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","0"],["7","-24","30","-16","3"],["-7","24","-30","16","-3"],["-16","48","-48","16","0"],["16","-48","48","-16","0"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["-22","20","-6"],["44","-40","12"],["46","-20","6"],["-92","40","-12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["-7","3","7","-3"],["14","-6","-14","6"],["16","0","-16","0"],["-32","0","32","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["8","0"],["0","0"],["-28","-12"],["0","0"],["20","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["7","-10","3"],["0","0","0"],["-23","26","-3"],["0","0","0"],["16","-16","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164698,"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-60496e4e2f616741"]},{"id":164699,"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-60496e4e2f616741"]},{"id":164700,"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-60496e4e2f616741"]}],"region_exact":{},"incoming_edges":[{"id":"e-60496e4e2f616741","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-8552357c29e7b09e","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":"a9b6705639fc59f7a851406ef1a269c566b83f44dfc8ea4f51a4135a85fb4588","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-984320118b653f6b","role":"sequence","name":"Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","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-03813340b1161041","kind":"fingerprint_related","node_a_id":"n-8552357c29e7b09e","node_b_id":"n-4d976aa636e25300","fingerprint":"f6-4e2cf4a8bda0ea40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e2cf4a8bda0ea40"},"other_id":"n-4d976aa636e25300","other":{"id":"n-4d976aa636e25300","name":"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":"f6-4e2cf4a8bda0ea40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["18","12"],["32","64","16"]],"card_rows":[["8"],["18","12"],["32","64","16"],["50","200","150","20"]],"row_sums_preview":["8","30","112","420","1584","6006","22880","87516"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u2299 C(n+1,k+1) (shift +2)","reverse"],"edges":["e-2d36970d4b706ae4","e-8b10d4d15ed56acc"],"text":"A007318 \u2192 \u2299 C(n+1,k+1) (shift +2) \u2192 reverse"},"headline":"A007318 \u2192 \u2299 C(n+1,k+1) (shift +2) \u2192 reverse","chain":null}},{"id":"l-4c3c81cc49fdba63","kind":"fingerprint_related","node_a_id":"n-c7016e67374634ec","node_b_id":"n-8552357c29e7b09e","fingerprint":"f6-4e2cf4a8bda0ea40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e2cf4a8bda0ea40"},"other_id":"n-c7016e67374634ec","other":{"id":"n-c7016e67374634ec","name":"reversal of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-4e2cf4a8bda0ea40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2","3"],["2","8","4"]],"card_rows":[["2"],["2","3"],["2","8","4"],["2","15","20","5"]],"row_sums_preview":["2","5","14","42","132","429","1430","4862"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["reverse"],"edges":["e-7ac651410315272d"],"text":"A108838 \u2192 reverse"},"headline":"A108838 \u2192 reverse","chain":null}},{"id":"l-53cf846d6f7fbe92","kind":"fingerprint_related","node_a_id":"n-82cc9c378f27e13e","node_b_id":"n-8552357c29e7b09e","fingerprint":"f6-4e2cf4a8bda0ea40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e2cf4a8bda0ea40"},"other_id":"n-82cc9c378f27e13e","other":{"id":"n-82cc9c378f27e13e","name":"derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-4e2cf4a8bda0ea40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["8","12"]],"card_rows":[["3"],["8","12"],["15","60","30"],["24","180","240","60"]],"row_sums_preview":["0","3","20","105","504","2310","10296","45045"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4ca874876022e136","steps":["D"],"edges":["e-7729d1773ce70fbd"],"text":"A062196 \u2192 D"},"headline":"A062196 \u2192 D","chain":null}},{"id":"l-7bdfa0f027913f27","kind":"fingerprint_related","node_a_id":"n-c963abfc0266d66c","node_b_id":"n-8552357c29e7b09e","fingerprint":"f6-4e2cf4a8bda0ea40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e2cf4a8bda0ea40"},"other_id":"n-c963abfc0266d66c","other":{"id":"n-c963abfc0266d66c","name":"Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-4e2cf4a8bda0ea40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["3","2"],["4","8","2"]],"card_rows":[["2"],["3","2"],["4","8","2"],["5","20","15","2"]],"row_sums_preview":["2","5","14","42","132","429","1430","4862"],"regions":["(-inf, +inf)","(-inf, 0)"],"region_exact":{},"oeis_ids":["A108838"],"derivation":{"root":"n-c963abfc0266d66c","steps":[],"edges":[],"text":"A108838"},"headline":"A108838: Triangle of Dyck paths counted by number of long interior inclines.","chain":null}},{"id":"l-944bd9b3e6a1a291","kind":"fingerprint_related","node_a_id":"n-7a4827a398e28fe4","node_b_id":"n-8552357c29e7b09e","fingerprint":"f6-4e2cf4a8bda0ea40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e2cf4a8bda0ea40"},"other_id":"n-7a4827a398e28fe4","other":{"id":"n-7a4827a398e28fe4","name":"Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-4e2cf4a8bda0ea40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["12","18"],["16","64","32"]],"card_rows":[["8"],["12","18"],["16","64","32"],["20","150","200","50"]],"row_sums_preview":["8","30","112","420","1584","6006","22880","87516"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A352880"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u2299 C(n+1,k+1) (shift +2)"],"edges":["e-2d36970d4b706ae4"],"text":"A007318 \u2192 \u2299 C(n+1,k+1) (shift +2)"},"headline":"A352880: Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","chain":"A007318 \u2192 \u2299 C(n+1,k+1) (shift +2)"}},{"id":"l-d4ec76d8958e2fd3","kind":"fingerprint_related","node_a_id":"n-28f1bce6fca8e347","node_b_id":"n-8552357c29e7b09e","fingerprint":"f6-4e2cf4a8bda0ea40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e2cf4a8bda0ea40"},"other_id":"n-28f1bce6fca8e347","other":{"id":"n-28f1bce6fca8e347","name":"derivative of narayana","first_index":1,"fingerprint":"f6-4e2cf4a8bda0ea40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","2"]],"card_rows":[["1"],["3","2"],["6","12","3"],["10","40","30","4"]],"row_sums_preview":["0","1","5","21","84","330","1287","5005"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["D"],"edges":["e-80d7d7190d4bcb89"],"text":"narayana \u2192 D"},"headline":"narayana \u2192 D","chain":null}},{"id":"l-ef8cea3275377558","kind":"fingerprint_related","node_a_id":"n-a889e9d86928450a","node_b_id":"n-8552357c29e7b09e","fingerprint":"f6-4e2cf4a8bda0ea40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e2cf4a8bda0ea40"},"other_id":"n-a889e9d86928450a","other":{"id":"n-a889e9d86928450a","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.","first_index":0,"fingerprint":"f6-4e2cf4a8bda0ea40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","2"]],"card_rows":[["1"],["3","2"],["6","12","3"],["10","40","30","4"]],"row_sums_preview":["0","1","5","21","84","330","1287","5005"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["D"],"edges":["e-d1059a276b20c64c"],"text":"A001263 \u2192 D"},"headline":"A001263 \u2192 D","chain":null}},{"id":"l-45ec24e265b15639","kind":"oeis_match","node_a_id":"n-8552357c29e7b09e","node_b_id":null,"fingerprint":null,"oeis_id":"A002740","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002740","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A002740","relation":"match_row_sums"}],"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:   2;\n  2:   9,     6;\n  3:  24,    48,    12;\n  4:  50,   200,   150,     20;\n  5:  90,   600,   900,    360,     30;\n  6: 147,  1470,  3675,   2940,    735,     42;\n  7: 224,  3136, 11760,  15680,   7840,   1344,    56;\n  8: 324,  6048, 31752,  63504,  52920,  18144,  2268,   72;\n  9: 450, 10800, 75600, 211680, 264600, 151200, 37800, 3600, 90;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4+2*n^3-10*n^2+4*n)*P(n,x) = -(52*n^4-52*n^3-22*n^2+22*n-(108*n^4-108*n^3-54*n^2+54*n)*x+(60*n^4-60*n^3-42*n^2+42*n)*x^2-(4*n^4-4*n^3-10*n^2+10*n)*x^3)*P(n-3,x) + (16*n^4-16*n^3-7*n^2+7*n-(48*n^4-48*n^3-24*n^2+24*n)*x+(48*n^4-48*n^3-30*n^2+30*n)*x^2-(16*n^4-16*n^3-16*n^2+16*n)*x^3-(3*n^2-3*n)*x^4)*d/dx P(n-3,x) - (92*n^4-46*n^3-44*n^2+22*n-(40*n^4-20*n^3-40*n^2+20*n)*x+(12*n^4-6*n^3-12*n^2+6*n)*x^2)*P(n-2,x) - (32*n^4-16*n^3-14*n^2+7*n+(6*n^2-3*n)*x-(32*n^4-16*n^3-14*n^2+7*n)*x^2-(6*n^2-3*n)*x^3)*d/dx P(n-2,x) + (20*n^4-28*n^2+8+(12*n^4-12*n^2)*x)*P(n-1,x) + (16*n^4-23*n^2+7-(16*n^4-26*n^2+10)*x-(3*n^2-3)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 9 + 6*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 9 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(52*n^4 - 52*n^3 - 22*n^2 + 22*n - (108*n^4 - 108*n^3 - 54*n^2 + 54*n)*x + (60*n^4 - 60*n^3 - 42*n^2 + 42*n)*x^2 - (4*n^4 - 4*n^3 - 10*n^2 + 10*n)*x^3)*P[n - 3, x] + (16*n^4 - 16*n^3 - 7*n^2 + 7*n - (48*n^4 - 48*n^3 - 24*n^2 + 24*n)*x + (48*n^4 - 48*n^3 - 30*n^2 + 30*n)*x^2 - (16*n^4 - 16*n^3 - 16*n^2 + 16*n)*x^3 - (3*n^2 - 3*n)*x^4)*D[P[n - 3, x], x] - (92*n^4 - 46*n^3 - 44*n^2 + 22*n - (40*n^4 - 20*n^3 - 40*n^2 + 20*n)*x + (12*n^4 - 6*n^3 - 12*n^2 + 6*n)*x^2)*P[n - 2, x] - (32*n^4 - 16*n^3 - 14*n^2 + 7*n + (6*n^2 - 3*n)*x - (32*n^4 - 16*n^3 - 14*n^2 + 7*n)*x^2 - (6*n^2 - 3*n)*x^3)*D[P[n - 2, x], x] + (20*n^4 - 28*n^2 + 8 + (12*n^4 - 12*n^2)*x)*P[n - 1, x] + (16*n^4 - 23*n^2 + 7 - (16*n^4 - 26*n^2 + 10)*x - (3*n^2 - 3)*x^2)*D[P[n - 1, x], x])/(4*n^4 + 2*n^3 - 10*n^2 + 4*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 9 + 6*x; for(n = 3, N, my(i = n + 1); v[i] = (-(52*n^4 - 52*n^3 - 22*n^2 + 22*n - (108*n^4 - 108*n^3 - 54*n^2 + 54*n)*x + (60*n^4 - 60*n^3 - 42*n^2 + 42*n)*x^2 - (4*n^4 - 4*n^3 - 10*n^2 + 10*n)*x^3)*v[i - 3] + (16*n^4 - 16*n^3 - 7*n^2 + 7*n - (48*n^4 - 48*n^3 - 24*n^2 + 24*n)*x + (48*n^4 - 48*n^3 - 30*n^2 + 30*n)*x^2 - (16*n^4 - 16*n^3 - 16*n^2 + 16*n)*x^3 - (3*n^2 - 3*n)*x^4)*deriv(v[i - 3], x) - (92*n^4 - 46*n^3 - 44*n^2 + 22*n - (40*n^4 - 20*n^3 - 40*n^2 + 20*n)*x + (12*n^4 - 6*n^3 - 12*n^2 + 6*n)*x^2)*v[i - 2] - (32*n^4 - 16*n^3 - 14*n^2 + 7*n + (6*n^2 - 3*n)*x - (32*n^4 - 16*n^3 - 14*n^2 + 7*n)*x^2 - (6*n^2 - 3*n)*x^3)*deriv(v[i - 2], x) + (20*n^4 - 28*n^2 + 8 + (12*n^4 - 12*n^2)*x)*v[i - 1] + (16*n^4 - 23*n^2 + 7 - (16*n^4 - 26*n^2 + 10)*x - (3*n^2 - 3)*x^2)*deriv(v[i - 1], x))/(4*n^4 + 2*n^3 - 10*n^2 + 4*n)); 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-984320118b653f6b","steps":["D"],"edges":["e-60496e4e2f616741"],"text":"A132812 \u2192 D"},"headline":"A132812 \u2192 D","chain":null}